On multifold packings of radius-1 balls in Hamming graphs

TL;DR

This paper investigates multifold packings of radius-1 balls in Hamming graphs, providing bounds, classifications, and properties of such packings, with implications for error-correcting and list-decodable codes.

Contribution

It introduces asymptotic bounds, characterizes optimal packings, and classifies small cases, advancing understanding of multifold packings in Hamming spaces.

Findings

Asymptotic bounds for maximum size of 2-fold 1-packings as alphabet size grows

Identification of optimal packings related to MDS codes when q ≥ 2n

Classification of all optimal binary 2-fold 1-packings up to length 9

Abstract

A $\lambda$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $\lambda$ times. The well-known $r$-error-correcting codes correspond to the case $\lambda=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an upper bound for the size of a binary $\lambda$-fold $1$-packing and a lower bound for the size of a binary multiple radius-$1$ covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary $2$-fold $1$-packings up to length $9$, in particular, establish the maximum size $96$ of a binary $2$-fold $1$-packing of length $9$; (e) prove some properties of $1$-perfect unitrades, which are a special case of $2$-fold $1$-packings. Keywords: Hamming graph, multifold ball packings, two-fold ball packings, list decodable codes, multiple coverings, completely regular codes, linear programming bound

Tables

Table 1. TABLE I: Numbers N ​ ( n ) 𝑁 𝑛 N(n) and N ∗ ​ ( n + 1 ) superscript 𝑁 𝑛 1 N^{*}(n+1) of inequivalent 2 2 2 -fold 1 1 1 -packings in H ​ ( n , 2 ) 𝐻 𝑛 2 H(n,2) and even-weight 2 2 2 -fold 1 1 1 -packings in H ​ ( n + 1 , 2 ) 𝐻 𝑛 1 2 H(n+1,2) , respectively, n ≤ 9 𝑛 9 n\leq 9 .

Equations

$$P_q^{\lambda }(n,r)\le \lfloor \lambda q^n/|B_r|\rfloor$$

$$\overline{C}=\{(x_1,\dots ,x_n,x_1+...+x_n)\mid (x_1,\dots ,x_n)\in C\}.$$

$$q^{n-1-o(1)}\le P_q^2(n,1)=o(q^{n-1}),$$

$$m^{\frac{3n-v+1}{2}-o(1)}\le f_n(m,v,3)=o(m^{\frac{3n-v+1}{2}})$$

$$q^{n-1}\le P_q^n(n,1)\le \frac{q^n}{q-1+1/n}.$$

$$\log P_{q}^{\lambda}(n,1)\simeq(n-1)\log q\qquad\mbox{as $q\to\infty$ and $n=\mathrm{const}$.}$$

$$\Vert \bar{v}{\Vert }^2\le abN,$$

$$\Vert F{\Vert }^2=\Vert {\chi }_C{\Vert }^2-\Vert \rho \overline{1}{\Vert }^2=\rho (1-\rho )|V|.$$

$$-r\rho \le (MF{)}_x\le \lambda -r\rho$$

$$\Vert MF{\Vert }^2\le r\rho (\lambda -r\rho )|V|.$$

$$\Vert MF{\Vert }^2={\theta }_0^2\Vert f_0{\Vert }^2+\dots +{\theta }_D^2\Vert f_D{\Vert }^2\ge {\alpha }^2(\Vert f_0{\Vert }^2+\dots +\Vert f_D{\Vert }^2)={\alpha }^2\Vert F{\Vert }^2.$$

$${\alpha }^2\rho (1-\rho )|V|\stackrel{\text{eqref}eq:FV}{=}{\alpha }^2\Vert F{\Vert }^2\stackrel{\text{eqref}eq:MFF}{\le }\Vert MF{\Vert }^2\stackrel{\text{eqref}eq:ab}{\le }r\rho (\lambda -r\rho )|V|.$$

$$\frac{|C|}{q^n}\le \frac{n^2(q-1)-n^2}{n^2(q-1{)}^2-n^2}=\frac{(q-1)-1}{(q-1{)}^2-1}=\frac{1}{q}.$$

$$C=\{(x_1,\dots ,x_n)\in \{0,\dots ,q-1{\}}^n\mid x_1+\dots +x_n=0\mathrm{mod}q\},$$

$$|C|B_k^{\prime }$$

$$2^n{B}_k$$

$$K_k(i)=\sum _{j=0}^k(-1{)}^j(\genfrac{}{}{0ex}{}{i}{j}\left)\right(\genfrac{}{}{0ex}{}{n-i}{k-j})$$

$$K_0(i)$$

$$K_{n-1}(i)$$

$$B_k^{\prime }=B_{n-k}^{\prime }.$$

$$\alpha (i)=\left(n-2i-2+(-1{)}^i\right)\left(n-2i+2+(-1{)}^i\right).$$

$$\alpha (i)$$

$$(n-3)B_0+2B_2\le \lambda n-4+\sigma ,$$

$$(n-3)A_0(x)+2A_2(x)\le n-3+n(\lambda -1)+\sigma -1=\lambda n-4+\sigma .$$

$$(n-3)A_0(x)+2A_2(x)$$

$$2\alpha (0)B_0^{\prime }$$

$$|C|\le \frac{2^n(\lambda n+2\lambda -4+\sigma )}{2\alpha (0)B_0^{\prime }}=\frac{2^{n-1}(\lambda n+2\lambda -4+\sigma )}{(n-1)(n+3)}.$$

$$\beta (i)=(n-2i{)}^2-2-2(-1{)}^i.$$

$$\beta (i)=0,\text{mbox}ifi\in \{n/2-1,n/2,n/2+1\};$$

$$(n-2)B_0+2B_2\le \lambda n-2$$

$$2\beta (0)B_0^{\prime }$$

Full text

I Introduction

It is well known that the error-correcting codes in Hamming spaces can be treated as packings of balls of fixed radius $r$. The balls in such packing do not intersect, every vertex of the space belongs to at most one of the balls, and this allows to correct an error of weight at most $r$ if we know that the transmitted information was represented by a codeword. One of the main problems in the area is to find the maximum number of balls in such packings, given the ball radius and the parameters of the metric space (for Hamming spaces, the alphabet size $q$ and the word length $n$). We consider the generalization of this problem to the so-called $\lambda$-fold packings. For such packing, every vertex of the space belongs to at most $\lambda$ balls of the packing. In coding theory, the corresponding codes are known as list decodable codes (for a good survey in the topic, see the introduction in [20]) and can also be used in information transmission, for example, in channels with noiseless feedback, see [1]. One of the main advantages of $\lambda$-fold packings is that the cardinality of such a packing can be essentially larger than $\lambda$ times the cardinality of the largest one-fold packing in the same space. In the current paper, we prove several bounds on the size of $\lambda$-fold packings of radius-$1$ balls in binary and $q$-ary Hamming schemes and consider properties of some special two-fold packings. **footnotetext: The work was funded by the Russian Science Foundation under grants 14-11-00555 and 18-11-00136. The results of the paper were presented in part at the IEEE International Symposium on Information Theory ISIT 2019, July 7–12, 2019, Paris.

The Hamming distance $\mathrm{d_{H}}(x,y)$ between two words $x$ and $y$ of the same length is the number of coordinates in which $x$ and $y$ differ. The Hamming graph (if $q=2$, the $n$-cube) $H(n,q)$ is a graph whose vertices are the words of length $n$ over the $q$-ary alphabet $\{0,\ldots,q-1\}$, two words being adjacent if and only if they differ in exactly one position. The weight $\mathrm{wt}(x)$ of a word $x$ is the number of nonzeros in $x$. The halved $n$-cube $\frac{1}{2}\mathrm{H}(n,2)$ is a graph whose vertices are the even-weight (or odd-weight) binary $n$-words, two words being adjacent if and only if they differ in exactly two positions. Two vertex sets of $H(n,q)$ are called equivalent if some isomorphism of $H(n,q)$ sends one of the sets to the other.

We will say that a multiset $C$ of vertices of $H(n,q)$ is an $\lambda$-fold $r$-packing (of length $n$) if for every vertex $x$ of $H(n,q)$ the number of elements of $C$ at distance at most $r$ from $x$ does not exceed $\lambda$. The concept of $1$-fold $r$-packing coincides with the well-known concept of $r$-error-correcting code. The sphere-packing bound for error-correcting codes is generalized to the obvious bound

[TABLE]

on the maximum cardinality $P_{q}^{\lambda}(n,r)$ of a $\lambda$-fold $r$-packing in $H(n,q)$, where $B_{r}$ is a radius-$r$ ball in $H(n,q)$.

A packing is called simple if it has no multiple elements and can be treated as an ordinary set, without multiplicities. In the literature, the simple $\lambda$-fold $r$-packings are also known as the ${\leq}\lambda$-list decodable codes with radius $r$, see e.g. [9, 2]. The only reason why we consider multisets is seeking generality: some of our results do not require the simplicity of considered packings.

Preceding results on the bounds on the size of a $\lambda$-fold $r$-packing (${\leq}\lambda$-list decodable code) were mainly focused on the asymptotics with growing packing radius. Blinovsky [9, 10] proved that there exists a sharp bound $\tau(\lambda,q)$ such that if $r=\tau n$, $\tau<\tau(\lambda,q)$ then the maximum cardinality of a $\lambda$-fold $r$-packing is exponentially large in $n$; in particular, $\tau(\lambda,2)=\frac{1}{2}-\frac{\binom{2k}{k}}{2^{2k+1}}$ where $k=\lfloor\frac{\lambda}{2}\rfloor$. In the case $\tau>\tau(\lambda,q)$, the maximum size of such packings is bounded by constant as $n\rightarrow\infty$ (see [11]). In [9], he also proved the asymptotic upper bound $\frac{\log|C|}{n}$ on the rate of a binary $\lambda$-fold $(\tau n)$-packings $C$. This bound was improved by Ashikhmin, Barg, and Litsyn [4] for $\lambda=2$ and by Polyanskiy [42] for $\lambda\geq 3$. How large can $C$ be when $\tau$ is just above the threshold $\tau(\lambda,q)$? For ordinary error-correcting codes ($\lambda=1$), the classical result of Levenshtein [36] says that the well-known Plotkin bound is sharp, namely, the size of the maximum code with radius $(\tau(1,2)+\varepsilon)n=(\frac{1}{4}+\varepsilon)n$ is $\frac{1}{4\varepsilon}+O(1)$ as $\varepsilon\rightarrow 0$. Alon et al. [2] proved that the maximum possible size of a $2$-fold packing with radius $(\tau(2,2)+\varepsilon)n=(\frac{1}{4}+\varepsilon)n$ is $\Theta(\frac{1}{\varepsilon^{3/2}})$. For $\lambda$-fold packings with radius $(\tau(\lambda,2)+\varepsilon)n$, there is an upper bound $O(\frac{1}{\varepsilon})$ for the size of $C$ as $\lambda$ is odd [2]. In all mentioned asymptotic results, the length and the packing radius grow, while the alphabet is fixed.

We are interested in other asymptotics and bounds for the size of optimal multifold packings, where the packing radius $r$ is fixed and the length or the size of the alphabet (or both) grows. In this paper, we focus on the case $r=1$, and obtain several different results, separately considering the case of arbitrary alphabet (Sections II and III) with growing alphabet size and the binary case (Sections IV–VI). Special attention is payed to studying so-called $1$-perfect unitrades, which are a special case of two-fold $1$-packings (where no radius-$1$ ball contains exactly one codeword). The study of these objects is motivated by the connection with the class of $1$-error-correcting perfect codes and by the fact that the classification of $1$-perfect unitrades of small parameters results in finding optimal binary two-fold $1$-packing. Further studying $1$-perfect unitrades with theoretical and computer-aided tools can result if finding more good (may be even optimal) two-fold packings, not only in the binary case.

In Section II we propose asymptotic bounds for $2$-fold $1$-packing in $H(n,q)$ as $q$ grows. In Section III we show that if $q\geq 2n$ then the maximum $n$-fold packings has size $q^{n-1}$; an example of such packing is a distance-$2$ MDS code. In Section IV, based on linear programming, we derive upper bounds on the size of a $\lambda$-fold $1$-packing in $H(n,2)$; as a corollary, we also establish corresponding lower bounds on the size of a $\mu$-fold $1$-covering. In Section V, we describe the optimal binary $2$-fold $1$-packings of length up to $9$ and their connection with completely regular codes and discuss multifold packings attaining the sphere-packing bound. In Section VI, we consider properties of $1$-perfect unitrades; we estimate the minimum cardinality and the inner radius of a $1$-perfect unitrade, show that all $1$-perfect unitrades up to the length $7$ are covered by systematic constructions, and classify the $1$-perfect unitrades of length $9$ (to be exact, their length-$10$ extensions), which results in finding an optimal $2$-fold $1$-packing of length $9$ and size $96$.

In the binary case, Sections IV–VI are focused on, it is often convenient to study the even-weight extensions of the $\lambda$-fold $1$-packings instead of the original objects. By this reason, in the end of the introduction we mention a fundamental one-to-one correspondence, which generalizes the well-known correspondence between binary $r$-error-correcting codes and their extended versions. A set of words is called even-weight if each of its elements has even weight. Given a multiset $C$ of binary words, its extension $\overline{C}$ is obtained from $C$ by appending the parity-check bit to all words:

[TABLE]

The inverse operation, projection, or puncturing, is removing the last symbol from all words of $C$. Another important operation, shortening, consists of removing from $C$ the words whose last symbol is different from [math] and then removing the last symbol [math] from all remaining words.

Proposition 1**.**

A multiset $C$ of binary words is a $\lambda$-fold $r$-packing of length $n$ if and only if its extension $\overline{C}$ is an even-weight $\lambda$-fold $r$-packing of length $n+1$.

Proof:

It is easy to check that if every radius-$r$ ball in $H(n,2)$ contains at most $\lambda$ words from $C$, then every radius-$r$ ball in $H(n+1,2)$ contains at most $\lambda$ words from $\overline{C}$, and if some radius-$r$ ball in $H(n,2)$ contains more than $\lambda$ words from $C$, then some radius-$r$ ball in $H(n+1,2)$ contains more than $\lambda$ words from $\overline{C}$. ∎

II Two-fold $1$-packings in $q$-ary Hamming graph

In this section, we estimate the asymptotic of the maximum size of a two-fold $1$-packing in $H(n,q)$ as $q$ grows with constant $n$. Then, we note that this also provides an estimation for the asymptotic of the maximum size of a $\lambda$-fold $1$-packing for every $\lambda$ between $2$ and $n$.

Theorem 1**.**

The maximum size $P_{q}^{2}(n,1)$ of a $2$-fold $1$-packing in $H(n,q)$, $n\geq 3$, satisfies

[TABLE]

as $q\rightarrow\infty$ and $n=\mathrm{const}$.

The case $n=2$ is not covered by this theorem, but considered in Corollary 1, see the next section: $P_{q}^{2}(2,1)=q$. The case $n=1$ is trivial: $P_{q}^{2}(1,1)=2$.

Proof:

An $n$-uniform hypergraph is a pair $(V,E)$ from a set $V$, whose elements are called vertices, and a set $E$ of $n$-subsets of $V$, called edges. Let $f_{n}(m,v,e)$ be the maximum number of edges in an $n$-uniform hypergraph on $m$ vertices such that the union of any $e$ edges has more than $v$ vertices.

In [3], it was shown that

[TABLE]

if $2\leq\frac{3n-v+1}{2}<n$ (for $(n,v)=(3,6)$, it was proved earlier in [47]).

Lower bound. Let us consider an $n$-uniform hypergraph with $q$ vertices and $f_{n}(q,n+\lambda+1,\lambda+1)$ edges such that the union of any $\lambda+1$ edges has more than $n+\lambda+1$ vertices. From every edge, we construct a codeword of length $n$ by listing the elements of the edge in some order (the order itself is not important, one can take random, or lexicographic, or any else). We state that the resulting code $C$ of size $f_{n}(q,n+\lambda+1,\lambda+1)$ is a $\lambda$-fold packing. Indeed, if it is not, then there are $\lambda+1$ codewords $x_{0}$, …, $x_{\lambda}$ at distance at most $1$ from some word $c$, the center of a ball. But $x_{0}$, …, $x_{\lambda}$ contain at most $n+\lambda+1$ different symbols in total ($n$ symbols of $c$, plus at most one unique symbol in each of $x_{i}$, $i=0,\ldots,\lambda$), which contradicts the definition of $f_{n}$. So, in the case $\lambda=2$ we see that the lower bound in (2) follows from the lower bound in (3).

Upper bound. Given a $2$-fold $1$-packing $C$, we first construct its subset $C^{\prime}$, without multiple codewords, such that $|C^{\prime}|\geq|C|/2$ and there are no two adjacent codewords in $C^{\prime}$. We can always do so because, by the definition of a $2$-fold $1$-packing, every codeword of $C$ has at most one neighbor in $C$, while a codeword of multiplicity $2$ has no neighbors in $C$. Next, we define a hypergraph with $nq$ vertices $(i,a)$, $i\in\{1,\ldots,n\}$, $a\in\{0,\ldots,q-1\}$ and $|C^{\prime}|$ edges $\{(1,c_{1}),\ldots,(n,c_{n}):$ $(c_{1},\ldots,c_{n})\in C^{\prime}\}$.

(*) We claim that the union of every three edges, corresponding to some codewords $x$, $y$, $z$ of $C^{\prime}$, has at least $n+4$ vertices. The proof of the claim is divided into three cases.

• •

If the distance between some two of $x$, $y$, $z$ is larger than $3$, then (*) is trivial.

• •

If the distance between two of them, say $x$ and $y$, is $3$, then the union of the corresponding two edges has $n+3$ vertices. If the edge corresponding to $z$ contains another vertex, then the union has $n+4$ vertices and (*) holds. Otherwise, in each position $z$ coincides with $x$ or $y$. Since $x$ and $y$ differ in only three positions, we see that $z$ is at distance $2$ from one of them and at distance $1$ from the other, which contradicts to the definition of $C^{\prime}$.

• •

There are two possibilities for three words to be at distance $2$ from each other, without loss of generality. One is $000...0$, $110...0$, $220...0$, and (*) holds in this case. The other is $1000...0$, $0100...0$, $0010...0$; such three words belong to the same radius-$1$ ball (centered in $0...0$), and hence cannot lie in $C^{\prime}$ by the definition of a $2$-fold $1$-packing.

Finally, we see that the constructed hypergraph cannot have more than $f_{n}(nq,n+3,3)$ edges, by the definition of $f_{n}$. Since it has $|C^{\prime}|$ edges and $|C^{\prime}|\geq|C|/2$, from (3) we derive $|C|\leq 2f_{n}(nq,n+3,3)=o((nq)^{n-1})=o(q^{n-1})$ as $q\to\infty$ and $n=\mathrm{const}$. ∎

Similar asymptotic estimation for the case $\lambda=n$ is rather simple (in the next section, we find the exact value of $P_{q}^{n}(n,1)$ for $q\geq 2n$).

Proposition 2**.**

In the case $\lambda=n$, the maximum size $P_{q}^{\lambda}(n,1)$ of a $\lambda$-fold $1$-packing in $H(n,q)$ admits the following bounds:

[TABLE]

Proof:

An example of $n$-fold $1$-packing is the distance-$2$ MDS code $\{(x_{1},\ldots,x_{n})\,:\,x_{1}+\ldots+x_{n}\equiv 0\bmod q\}$. Such code exists for any $n$ and $q$ and its cardinality equals $q^{n-1}$.

Since $|B_{1}|=n(q-1)+1$, we conclude from (1) that $|C|(n(q-1)+1)\leq nq^{n}$. ∎

For an arbitrary $n$, the largest $\lambda$-fold $1$-packing, $\lambda=2,\ldots,n$, has the cardinality between the cardinalities of largest $2$-fold and $n$-fold $1$-packings. Thus, we have the following.

Theorem 2**.**

If $P_{q}^{\lambda}(n,1)$ is the maximum cardinality of a $\lambda$-fold $1$-packing in $H(n,q)$, $\lambda\in\{2,\ldots,n\}$, then

[TABLE]

Another observation which can be made from Theorem 1 and Proposition 2 is that $P_{q}^{2}(n,1)=o(q^{n-1})$ while $P_{q}^{n}(n,1)=\Omega(q^{n-1})$, as a function in $q$. So, the answer to the following question meets $2<\lambda_{n}\leq n$.

Problem 1**.**

What is the value $\lambda_{n}$ such that $P_{q}^{\lambda_{n}-1}(n,1)=o(q^{n-1})$ but $P_{q}^{\lambda_{n}}(n,1)=\Omega(q^{n-1})$?

Brown, Erdős, and Sós [15] conjectured that $f_{n}(q,l(n-k)+k+1,l)=o(q^{k})$ as $q\rightarrow\infty$ and $2\leq k<n$, $l>2$. In the special case $k=n-1$, their conjecture implies that $f_{n}(q,n+l,l)=o(q^{n-1})$ as $q\rightarrow\infty$ and $l>2$. The arguments in the paragraph “Lower bound” of the proof of Theorem 1 (where $l=\lambda+1$) shows that the conjecture is true for every $l\leq\lambda_{n}$; so, finding $\lambda_{n}$ implies solving the conjecture for all smaller values of $l$. However, the inverse connection is not so clear, as the “Upper bound” arguments in the proof of Theorem 1 are not generalized to an arbitrary $\lambda$.

III Distance-$2$ MDS codes are optimal

A distance-$d$ MDS code is a set of vertices of $H(n,q)$ with cardinality $q^{n-d+1}$ and minimum distance $d$ between codewords. Distance-$2$ MDS codes exist for any $n,q\geq 2$, for example, $\{(x_{1},\ldots,x_{n})\,:\,x_{1}+\ldots+x_{n}\equiv 0\bmod q\}$. As can be seen from Proposition 2, the distance-$2$ MDS codes are asymptotically optimal $n$-fold $1$-packings as $q$ grows. In this section, we prove a general upper bound on the size of a $\lambda$-fold $1$-packing in any regular graph, which in particular shows that the distance-$2$ MDS codes are indeed largest $n$-fold $1$-packings if $q\geq 2n$. We conjecture that this also holds for $q\geq n$.

Theorem 3**.**

Let $G=(V,E)$ be a $r$-regular graph (or multigraph, with multiple edges and/or loops), and let $\alpha$ be a nonnegative constant such that $|\theta|\geq\alpha$ for any eigenvalue $\theta$ of $G$. If $C$ is a multiset of vertices of $G$ such that every vertex from $V$ is adjacent to at most $\lambda$ elements of $C$ (taking into account the multiplicity of the edges in the case of multigraph), then $\displaystyle\frac{|C|}{|V|}\leq\frac{r\lambda-\alpha^{2}}{r^{2}-\alpha^{2}}$.

Before proving the theorem, we derive one simple inequality.

Lemma 1**.**

Assume that a vector $\bar{v}=(v_{1},\ldots,v_{N})$ from $\mathbb{R}^{N}$ is orthogonal to the all-one vector ${\overline{1}}$ (that is, $v_{1}+\ldots+v_{N}=0$). If $a$ and $b$ are nonnegative constants such that $-a\leq v_{i}\leq b$ for every $i$ from $\{1,\ldots,N\}$, then

[TABLE]

where $\|\bar{v}\|^{2}=v_{1}^{2}+\ldots+v_{N}^{2}$; moreover, $\|\bar{v}\|^{2}=abN$ if and only if $v_{i}\in\{-a,b\}$, $i=1,\ldots,N$.

Proof:

Consider the vector $\bar{u}=(u_{1},\ldots,u_{N})=\bar{v}+\frac{a-b}{2}{\overline{1}}$. Straightforwardly, $|u_{i}|\leq\frac{a+b}{2}$ for $i=1,\ldots,N$. Consequently, $\|\bar{u}\|^{2}\leq\frac{(a+b)^{2}N}{4}$, with equality only if all components are $\pm\frac{a+b}{2}$. From the orthogonality of $\bar{v}$ and ${\overline{1}}$, we obtain $\|\bar{v}\|^{2}=\|\bar{u}\|^{2}-\|\frac{a-b}{2}{\overline{1}}\|^{2}\leq\frac{(a+b)^{2}N}{4}-\frac{(a-b)^{2}N}{4}=abN$, with equality only if all components of $\bar{v}$ are in $\{-a,b\}$. ∎

Proof:

We consider the space of functions $f:V\to\mathbb{R}$, treated as $|V|$-tuples whose coordinates are indexed by the vertices from $V$. The graph $G$ can be represented by the adjacency matrix $M$ of size $|V|\times|V|$ whose element $M_{a,b}$, $a,b\in V$, equals the multiplicity of the edge $\{a,b\}$ in the multigraph. The eigenvalues $\theta_{0},\ldots,\theta_{D}$ of the adjacency matrix are also known as the eigenvalues of the graph. By $\chi_{C}:V\to\{0,1,2,\ldots\}$, we denote the multiplicity vector of a vertex multiset $C$. In particular, $\overline{1}=\chi_{V}$. Denote $\rho=\frac{|C|}{|V|}$ and $F=\chi_{C}-\rho{\overline{1}}$. So, $\chi_{C}=\rho{\overline{1}}+F$, where $F\perp{\overline{1}}$. Consequently,

[TABLE]

Consider the vector $\bar{v}=(v_{x})_{x\in V}=M\chi_{C}=r\rho{\overline{1}}+MF$. Since, by the hypothesis of the theorem, every vertex $x$ from $V$ has at most $\lambda$ neighbors in $C$, we have $0\leq v_{x}\leq\lambda$, and hence

[TABLE]

(here the right expression is positive because $\rho\leq\lambda/(r+1)$ by the sphere-packing bound). From the regularity of the graph and $F\perp{\overline{1}}$, we find $(MF,\overline{1})=(F,M\overline{1})=r(F,\overline{1})=0$, where $(\cdot,\cdot)$ is the inner product. So, $MF\perp{\overline{1}}$, and by Lemma 1 we get

[TABLE]

Consider the representation $F=f_{0}+\ldots+f_{D}$ of $F$ as the sum of eigenvectors $f_{i}$ of $M$ corresponding to the eigenvalues $\theta_{i}$, $i=0,\ldots,D$. Since $MF=Mf_{0}+\ldots+Mf_{D}=\theta_{0}f_{0}+\ldots+\theta_{D}f_{D}$ and the vectors $f_{i}$ are pairwise orthogonal, we find

[TABLE]

Finally, we conclude that

[TABLE]

So, $\alpha^{2}(1-\rho)\leq r(\lambda-r\rho)$, and $\frac{|C|}{|V(G)|}=\rho\leq\frac{r\lambda-\alpha^{2}}{r^{2}-\alpha^{2}}$. ∎

Corollary 1**.**

If $q\geq 2n$, then the maximum cardinality $P_{q}^{n}(n,1)$ of an $n$-fold $1$-packing in $H(n,q)$ equals $q^{n-1}$ and every $n$-fold $1$-packing in $H(n,q)$ of cardinality $q^{n-1}$ is a distance-$2$ (MDS) code.

Proof:

The eigenvalues of $H(n,q)$ are $\theta_{i}=-n+qi$, $i=0,\ldots,n$ [14]. If $q\geq 2n$, then obviously $|\theta_{i}|\geq n$ for every $i$. The graph $H(n,q)$ is $n(q-1)$-regular, so by Theorem 3 every $n$-fold $1$-packing $C$ satisfies

[TABLE]

On the other hand, any distance-$2$ MDS code $C$ in $H(n,q)$, for example,

[TABLE]

is an $n$-fold $1$-packing that meets $\frac{|C|}{q^{n}}=\frac{1}{q}$.

It remains to show that every such packing $C$ is a distance-$2$ code. As follows from Lemma 1, (5) holds with equality only if every vertex has exactly [math] or $\lambda$ neighbors from $C$. But, by the definition of a $\lambda$-fold $1$-packing, any codeword of $C$ has at most $\lambda-1$ neighbors from $C$; hence, it has [math] neighbors from $C$. So, the minimum distance between different codewords is at least $2$. ∎

If $q=n-1$ and $q$ is a prime power, then for any $\lambda$ from $\{1,\ldots,q^{2}\}$ we can construct a $\lambda$-fold $1$-packing in $H(n,q)$ as the union of $\lambda$ cosets of the $1$-error-correcting Hamming code of cardinality $q^{n-2}$. Such packing has cardinality $\lambda q^{n-2}=\frac{\lambda q^{n}}{nq-q+1}$, which is larger than $\frac{\lambda q^{n}}{nq}$; in particular, if $\lambda=n$, then it is larger than the cardinality $q^{n-1}$ of a distance-$2$ MDS code.

Conjecture. If $q\geq n$, then the cardinality of the largest $n$-fold $1$-packing in $H(n,q)$ is $q^{n-1}$.

IV Bounds for multifold packings

and multiple coverings in the binary case

In this section, we prove upper bounds on the size of $\lambda$-fold $1$-packings in the $n$-cube $H(n,2)$, generalizing the bounds and approach from [7] for $1$-error-correcting codes, corresponding to the case $\lambda=1$. As a corollary, based on the connection between multifold packings and multiple coverings (see the definition in Subsection IV-B), new lower bounds for binary multiple radius-$1$ coverings are established.

IV-A Multiple $1$-packings

The weight distribution of a code $C$ of length $n$ is the sequence $\{A_{i}\}_{i=0}^{n}$, where $A_{i}$ is the number of the codewords of weight $i$ in $C$. The weight distribution $\{A_{i}(x)\}_{i=0}^{n}$ of $C$ with respect to a word $x$ is the weight distribution of the code $C+x$. The distance distribution $\{B_{i}\}_{i=0}^{n}$ of $C$ is defined as the average weight distribution of $C$ with respect to all its codewords: $B_{i}=\frac{1}{|C|}\sum_{x\in C}A_{i}(x)$. The main result of this section is the following bound.

Theorem 4**.**

The maximum size $P_{2}^{\lambda}(n,1)$ of a binary $\lambda$-fold $1$-packing of length $n$, where $\lambda\equiv\sigma\bmod 2$, $\sigma\in\{0,1\}$, satisfy

1. (a)

$\displaystyle P_{2}^{\lambda}(n,1)\leq\frac{2^{n}(\lambda n+3\lambda-4+\sigma)}{n(n+4)}$*   if $n\equiv 0\bmod 4$, * 2. (b)

$\displaystyle P_{2}^{\lambda}(n,1)\leq\frac{2^{n}(\lambda n+\lambda-2)}{(n-1)(n+3)}$*   if $n\equiv 1\bmod 4$, * 3. (c)

$\displaystyle P_{2}^{\lambda}(n,1)\leq\frac{2^{n}(\lambda n+\lambda-2+\sigma)}{n(n+2)}$*   if $n\equiv 2\bmod 4$, * 4. (d)

$\displaystyle P_{2}^{\lambda}(n,1)\leq\frac{2^{n}\lambda}{n+1}$*   if $n\equiv 3\bmod 4$.*

We will prove this theorem in the following form, whose claim is equivalent to the claim of Theorem 4 by Proposition 1.

Theorem 5**.**

Every even-weight $\lambda$-fold $1$-packing $C$ of length $n$, where $\lambda\equiv\sigma\bmod 2$, $\sigma\in\{0,1\}$, satisfy

1. (a)

$\displaystyle|C|\leq\frac{2^{n-1}(\lambda n+2\lambda-4+\sigma)}{(n-1)(n+3)}$*  if $n\equiv 1\bmod 4$, * 2. (b)

$\displaystyle|C|\leq\frac{2^{n-1}(\lambda n-2)}{(n-2)(n+2)}$*  if $n\equiv 2\bmod 4$, * 3. (c)

$\displaystyle|C|\leq\frac{2^{n-1}(\lambda n-2+\sigma)}{(n-1)(n+1)}$*  if $n\equiv 3\bmod 4$, * 4. (d)

$\displaystyle|C|\leq\frac{2^{n-1}\lambda}{n}$*  if $n\equiv 0\bmod 4$.*

Proof:

Let $\{B^{\prime}_{i}\}_{i=0}^{n}$ be the MacWilliams transform of the distance distribution $\{B_{i}\}_{i=0}^{n}$ of $C$; that is,

[TABLE]

where

[TABLE]

is a Krawtchouk polynomial; in particular,

[TABLE]

It is well known that $B^{\prime}_{0}=1$ and $B^{\prime}_{i}\geq 0$ for $1\leq i\leq n$ [19].

As $C$ is an even-distance code, $B_{i}=0$ for odd $i$, and, since $K_{n-k}(i)=(-1)^{i}K_{k}(i)$, we have

[TABLE]

(a) Let $n\equiv 1\bmod 4$. Define $\alpha(i)=(n-3)K_{0}(i)+2K_{2}(i)+2K_{n-1}(i)$. Direct calculations now show that

[TABLE]

It turns to $(n-3-2i)(n+1-2i)$ with zeros in $\frac{n-3}{2}$, $\frac{n+1}{2}$ if $i$ is odd, and to $(n-1-2i)(n+3-2i)$ with zeros in $\frac{n-1}{2}$, $\frac{n+3}{2}$ if $i$ is even. From (9) and $n\equiv 1\pmod{4}$ we derive

[TABLE]

From the packing condition we have $B_{n-1}\leq\lambda$ and, moreover,

[TABLE]

with equality only if there are no codewords of multiplicity more than $1$. (Indeed, for a vertex $x$ of multiplicity $A_{0}(x)=1$, the number $A_{2}(x)$ of codewords at distance $2$ from $x$ does not exceed $\lfloor n(\lambda-1)/2\rfloor=(n(\lambda-1)+\sigma-1)/2$; so,

[TABLE]

For a larger multiplicity, $A_{0}(x)\geq 2$, we have

[TABLE]

which is smaller than the right part of (11).)

Utilizing (8), we then get

[TABLE]

and thereby

[TABLE]

(b) Let $n\equiv 2\bmod 4$. Define $\beta(i)=(n-2)K_{0}(i)+2K_{2}(i)-2K_{n}(i)$. Straightforwardly,

[TABLE]

Since $n/2$ is odd, we see that

[TABLE]

for any other integer $i$, we have $\beta(i)>0$. From the packing condition we have

[TABLE]

with equality if and only if $B_{0}=1$ (i.e., there are no codewords of multiplicity more than $1$) and $B_{2}=(\lambda-1)n/2$. Then, we get

[TABLE]

and thereby

[TABLE]

(c) Let $n\equiv 3\bmod 4$. Define $\gamma(i)=(n-1)K_{0}(i)+2K_{2}(i)$. Straightforwardly,

[TABLE]

Obviously,

[TABLE]

and $\gamma(i)>0$ for any other integer $i$.

With an argument similar to that for (11), we have

[TABLE]

(the equality implies $B_{0}=1$ if $\sigma=1$, but for even $\lambda$ we cannot make this conclusion). Then, we get

[TABLE]

and hence

[TABLE]

(d) Let $n\equiv 0\bmod 4$. Define $\delta(i)=nK_{0}(i)+2K_{2}(i)$. Straightforwardly, $\delta(i)=(n-2i)^{2}\geq 0$, and $\delta(i)=0\Leftrightarrow i=n/2.$

From the $\lambda$-fold packing condition, we have

[TABLE]

(in this case, the equality does not required $B_{0}$ to be $1$, so multiple codewords are allowed). Then, we get

[TABLE]

and hence

[TABLE]

∎

Corollary 2**.**

Assume that $C$ is an even-weight $\lambda$-fold $1$-packing of length $n$, and assume that one of equations (a)–(d) in Theorem 5, in respect to $n\bmod 4$, is satisfied with equality. Then

• (i)

if $n\equiv 1,2\bmod 4$, or $n\equiv 3\bmod 4$ and $\lambda$ is odd, then $C$ is simple (there are no multiple codewords);

• (ii)

if $C$ is simple, then its weight distribution with respect to any codeword $c\in C$ is uniquely determined by the parameters $n$ and $\lambda$; in particular, there are exactly $\lfloor n(\lambda-1)/2\rfloor$ codewords at distance $2$ from $c$.

Proof:

Assume that $C$ is an even-weight $\lambda$-fold $1$-packing of length $n$, $n\equiv 1\bmod 4$, and assume that the inequality (a) in Theorem 5 is satisfied with equality. This means that we have equalities everywhere in (IV-A). As follows from (11) and the note after it, the equality in (IV-A) implies $B_{0}=1$ (which proves claim (i)), $B_{2}=\lfloor n(\lambda-1)/2\rfloor$, and $B_{n-1}=\lambda$. Since $A_{0}(x)\geq 1$, $A_{2}(x)\leq\lfloor n(\lambda-1)/2\rfloor$, and $A_{n-1}(x)\leq\lambda$ for every codeword $x$, we also have $A_{0}(x)=1$, $A_{2}(x)=\lfloor n(\lambda-1)/2\rfloor$, and $A_{n-1}(x)=\lambda$. Remind also that $B_{i}=A_{i}(x)=0$ for every odd $i$.

Next, consider the dual distance distribution $\{B^{\prime}_{i}\}_{i=0}^{n}$. From (10) and the equality in (IV-A) we find that $B^{\prime}_{i}=0$ for all $i$ except [math], $(n-3)/2$, $(n-1)/2$, $(n+1)/2$, $(n+3)/2$, $n$. Moreover, we know that $B^{\prime}_{i}=B^{\prime}_{n-i}$ for all $i$ and $B^{\prime}_{0}=B^{\prime}_{n}=1$. So, for complete determining $\{B^{\prime}_{i}\}_{i=0}^{n}$, it remains to know $B^{\prime}_{(n-3)/2}$ and $B^{\prime}_{(n-1)/2}$. These two values can be found from two equations (IV-A), $k=0,2$. So, the dual distance distribution and, hence, the distance distribution are uniquely determined.

The same arguments can be applied to the dual weight distribution $\{A^{\prime}_{i}(x)\}_{i=0}^{n}$ calculated from $\{A_{i}(x)\}_{i=0}^{n}$ by the same formulas as $\{B^{\prime}_{i}\}_{i=0}^{n}$ from $\{B_{i}\}_{i=0}^{n}$ (IV-A). Indeed, by [37, Theorem 7(b) in Ch.5, §5], $B^{\prime}_{i}=0$ implies $A^{\prime}_{i}(x)=0$. We also have $A^{\prime}_{i}(x)=A^{\prime}_{n-i}(x)$ and $A^{\prime}_{0}(x)=A^{\prime}_{n}(x)=1$, and we know $A_{0}(x)$ and $A_{2}(x)$. So, we can completely determine $\{A^{\prime}_{i}(x)\}_{i=0}^{n}$ and then $\{A_{i}(x)\}_{i=0}^{n}$.

For $n\equiv 2,3,0\bmod 4$, the proof is similar. ∎

IV-B Multiple coverings

A set $C$ of vertices of $H(n,q)$ is called a $q$-ary $(n,\cdot,r,\mu)$ multiple covering [17, Ch. 14] if for every vertex $x$ of $H(n,q)$ the number of elements of $C$ at distance at most $r$ from $x$ is not less than $\mu$. Trivially, the complement of any simple $\lambda$-fold $r$-packing in $H(n,q)$ is a $q$-ary $(n,\cdot,r,\mu)$ multiple covering, where $\mu=|B_{r}|-\lambda$, and vice versa. As a consequence, the following lower bounds on the size of binary radius-$1$ coverings are derived from Theorem 4.

Theorem 6**.**

The minimum cardinality $K(n,1,\mu)$ of a binary $(n,\cdot,1,\mu)$ covering, where $\mu\equiv\tau\bmod 2$, $\tau\in\{0,1\}$, satisfy

1. (a)

$\displaystyle K(n,1,\mu)\geq\frac{2^{n}(\mu n+3\mu+\tau)}{n(n+4)}$*   if $n\equiv 0\bmod 4$, * 2. (b)

$\displaystyle K(n,1,\mu)\geq\frac{2^{n}(\mu n+\mu-2)}{(n-1)(n+3)}$*   if $n\equiv 1\bmod 4$, * 3. (c)

$\displaystyle K(n,1,\mu)\geq\frac{2^{n}(\mu n+\mu+\tau)}{n(n+2)}$*   if $n\equiv 2\bmod 4$, * 4. (d)

$\displaystyle K(n,1,\mu)\geq\frac{2^{n}\mu}{n+1}$*   if $n\equiv 3\bmod 4$.*

These bounds update the previous lower bounds [25, 48] in the table [48, Table 1] of small values for $K(n,1,\mu)$ in the following positions.

[TABLE]

V Optimal packings

In this section, we discuss some optimal packings, namely, binary two-fold packings attaining the bound in Theorem 4 for small $n$ and multifold $1$-packings attaining the sphere-packing bound. Many of the discussed packings are related with very regular objects called equitable partitions.

A partition $\pi=(C_{0},C_{1},\ldots,C_{m})$ of the vertices of a graph (in our case, $H(10,2)$) is called equitable if for every $i$ and $j$ from $\{0,1,\ldots,m\}$ there is an integer $s_{i,j}$ such that each vertex $v$ in $C_{i}$ has exactly $s_{i,j}$ neighbors in $C_{j}$. The matrix $(s_{i,j})_{i,j=0}^{m}$ is called the intersection matrix (sometimes, the quotient matrix of the graph with respect to the partition). If it is tridiagonal (equivalently, $\pi$ is a distance partition with respect to $C_{0}$), then $C_{0}$ is called a completely regular code of covering radius $m$ with intersection matrix $(s_{i,j})_{i,j=0}^{m}$ (alternatively, the parameters of a completely regular code are often given in the form of the intersection array $(s_{0,1},\ldots,s_{m-1,m};s_{1,0},\ldots,s_{m,m-1})$), see e.g. the survey [13] for the background of this important concept.

V-A Optimal $2$-fold packings in $H(n,2)$ for small $n$

As special cases of Theorems 4 and 5, we have the bounds for $P_{2}^{n}(2,1)$ reflected in Table I.

As a result of the exhaustive computer-aided search (see the appendix for the description of the algorithm), we found that all these bounds for $n\leq 9$ are tight, and we found the number of all equivalence classes of simple optimal $2$-fold $1$-packings in $H(n,2)$, $n\leq 9$, and of their extensions in $H(n+1,2)$. It happens that each of packing with odd $n\leq 9$ from the classification, as well as each even-weight packing with any $n\leq 10$, can be represented as a cell of an equitable partition with special intersection matrix, depending on the parameters of the packing:

[TABLE]

[TABLE]

It is not difficult to see the inverse: the last cell of an equitable partition with one of the mentioned intersection matrices is an optimal (even/odd-weight) two-fold $1$-packing (the proportions between the cardinalities of the cells can be seen from the matrix, as well as the parity of each cell in the case of $S^{...}_{\mathrm{even}}$; the packing condition can be seen from the last column).

We refer the connection with equitable partitions as a computational result. However, for some parameters it can be explained theoretically with the technique developed in [29, 31], may be generalizing to packings attaining the bound with some other special values of $n$ and $\lambda$ (in [29, 31] it was shown for $\lambda=1$, $n=2^{m}-2$ and $n=2^{m}-3$, respectively, with corresponding intersection matrices). Here, we do not focus on the details of that theory, because the number of cases we are interested is finite, and the proof from [29, 31] works without essential changes for all considered parameters except the ones related with the matrix $S^{9}_{\mathrm{even}}$.

Another empiric fact is that every $2$-fold $1$-packing in $H(n,2)$, $n=2,4,6,8$, can always be uniquely represented as a shortened $2$-fold $1$-packing of length $n+1$ (similarly, for even-weight extensions). Again, we have theoretical explanations for some parameters, but not for all. Similar natural questions were previously considered in the case $\lambda=1$: every optimal $1$-packing in $(n,2)$, $n=2^{m}-2$, is always a shortened optimal $1$-packing in $(n+1,2)$ [8], while this is not always true for $n=2^{m}-3$ and $n=2^{m}-4$ [40, 34].

The most interesting $2$-fold $1$-packings from the considered classification up to length $9$ are the optimal packings in $H(9,2)$ and their length-$10$ extensions (indeed, the packings in $H(8,2)$ are obtained by shortening from the packings in $H(9,2)$; the size of the packings in $H(7,2)$ and $H(6,2)$ is not better than the union of two cosets of a $1$-error-correcting code; the extention of the best packing in $H(6,2)$ is equivalent to a unique $2$-$(6,3,2)$-design; $H(5,2)$ is again a shortened case; the other are trivial). The cardinality $96$ of these packings is very close to the sphere-packing bound $\lfloor 2\cdot 2^{9}/(1+9)\rfloor=102$. To compare, the largest $1$-error-correcting code in $H(9,2)$ has $40$ codewords [6], and the union of two disjoint such codes is a $2$-fold $1$-packing of cardinality $80$ only. The intersection matrix $S_{\mathrm{even}}^{10}$ of the equitable partition corresponding to the extended packings is not tridiagonal, but unifying the last two cells (which are in fact an even-weight $2$-fold packing $P$ of size $96$ and its antipode $P+1111111111$) results in an equitable partition with tridiagonal intersection matrix. So, the first cell of the equitable partition is a completely regular code with intersection array $(10,9,4;1,6,10)$. One such code, the only linear one, was already known [45, Theorem 1(2)]; some details about all three nonequivalent completely regular codes, including their propelinear structures, and corresponding optimal $2$-fold packings can be found in the proceedings [35] (those details were considered to be too peculiar to include them in the current paper). Here, we present only one example of an optimal even-weight $2$-fold $1$-packing of size $96$:

[TABLE]

Remark 1*.*

We note that our classification results do not imply the nonexistence of other length-$10$ completely regular codes with intersection array $(10,9,4;1,6,10)$. However, this can be easily derived from the following known facts: (i) if such code contains the all-zero word, then there are $15$ weight-$4$ codewords and they form a $2$-design, see [24, Th. 8]; (ii) any completely regular code $C$ with considered parameters is self-complementary, that is, $C=C+1111111111$, see, e.g., [12]; (iii) there are exactly $3$ isomorphism classes of $2$-$(10,4,2)$ designs, see [38, Table 1.25]. So, by some “magic” reason, the set of vertices at maximum distance from each binary length-$10$ completely regular code with intersection array $(10,9,4;1,6,10)$ can be splitted into two $2$-fold $1$-packings, in more than one way, and all ways to splits result in equivalent packings, for the same code.

V-B Multifold perfect codes

By analogy with classical perfect codes, attaining the sphere-packing bound, a $\lambda$-fold $r$-packings of cardinality $\lambda q^{n}/|B_{r}|$ (see (1)) in $H(n,q)$ can be called perfect, or alternatively, a $\lambda$-fold $r$-perfect code. Moreover, such packings are at the same time optimal $\lambda$-fold coverings, and they also known as perfect multiple coverings [17, Section 14.2]. The existence of multiple coverings with radius larger than $1$ is a complicate problem, especially for small $\lambda$. A survey can be found in [17, Section 14.2]; in [52], Vorob’ev considered multifold $r$-perfect codes in $H(n,2)$, $r>1$, related with equitable $2$-partitions. In the rest of this section, we focus on the case $r=1$. For a $\lambda$-fold $1$-pefrect code $C$, every radius-$1$ ball in $H(n,q)$ contains exactly $\lambda$ codewords. Moreover, if $r=1$ and $C$ has no multiple codewords, then $C$ and its complement form an equitable partition with intersection matrix

[TABLE]

The famous Lloyd condition (see, e.g., [23, Theorem 9.3.3] for equitable partitions) says that the eigenvalue $-1$ of this intersection matrix must belong to the eigenspectrum $\{-n+iq\mid i\in\{0,\ldots,n\}\}$ of the Hamming graph $H(n,q)$, which is equivalent to $n\equiv 1\bmod q$ (for example, this shows that a $5$-fold $1$-perfect code of size $32$ in $H(3,4)$ does not exist). Existence of such partitions in the binary case was shown in [22], as a part of a general theory of equitable $2$-partitions of $H(n,2)$ (which was partially generalized to an arbitrary $q$ in a recent work [5]), but the sufficiency of the sphere-packing condition for the existence of multifold $1$-perfect codes was known earlier for any prime $q$:

Proposition 3** ([17, Theorem 14.2.4]).**

Assume that $q$ is prime. Simple (without multiple codewords) $\lambda$-fold $1$-perfect codes in $H(n,q)$ exist if and only if $K=\lambda q^{n}/|B_{1}|$ is integer and $\lambda\leq|B_{1}|$, where $B_{1}$ is a radius-$1$ ball in $H(n,q)$, $|B_{1}|=(q-1)n+1$.

The sufficiency in Proposition 3 is proved by constructing a code as the union of $\kappa$ cosets of a linear $\lambda/\kappa$-fold $1$-perfect code of dimension $k$ (defined, for example, by a parity-check matrix, generalizing the Hamming code (see, e.g., [37, p.193])), where

[TABLE]

The same argument works to construct $\lambda$-fold $1$-perfect codes in $H(n,q)$ for any prime-power $q$; so, (18) is sufficient for the existence, but not necessary: if $q$ is not prime, then not all integer $K$ are representable in the form (18). The smallest example (also satisfying the Lloyd condition) is $q=4$, $n=13$, $\lambda=5$. In this case, $K=2^{23}=2\cdot 4^{11}$, and we see that $\lambda/\kappa$ is not integer for any representetion of $K$ in the form $\kappa 4^{k}$. So, a code cannot be constructed as the union of cosets of a linear multifold $1$-perfect code. However, we can construct a $5$-fold $1$-perfect code as an additive code: we first construct a binary linear code as the kernel of the check matrix

[TABLE]

then treat binary words of length $26$ as quaternary words of length $13$ in the alphabet $\{00$, $01$, $10$, $11\}$. To make sure that the code is $5$-fold $1$-perfect, one can check that each of $8$ different syndromes (height-$3$ binary columns) is represented in exactly $5$ ways as $He^{\mathrm{T}}$ where $e$ is one of $40$ words of weight at most $1$ (in the metric of $H(13,4)$): $00...00$, $\underline{01}00...00$, $\underline{10}00...00$, $\underline{11}00...00$, $00\underline{01}00...00$, …, $00...00\underline{11}$.

Problem 2**.**

Is the integrality of $\frac{\lambda q^{n}}{(q-1)n+1}$ sufficient for the existence of $\lambda$-fold $1$-perfect codes in $H(n,q)$, where $q$ is a prime power but not prime?

If $q$ is not a prime power, then the problem of the existence of nontrivial multifold $1$-perfect codes is open, generalizing the famous problem of the existence of perfect codes. Up to our knowledge, no notrivial multifold $1$-perfect codes are known over a none-prime-power alphabet.

In the binary case, $\lambda$-fold $1$-perfect codes attain the bound of Theorem 4(d), which trivially coincides with the sphere-packing bound; moreover, after shortening, $2$-fold $1$-perfect codes turn to packings attaining the bound of Theorem 4(c). This observation exhaust the known infinite series of $\lambda$-fold $1$-packings attaining the bounds of Theorem 4 (Theorem 5, for extended packings) with $1<\lambda\leq n$. Moreover, together with the results of Section V-A, it cover all possible $2$-fold packings lying on those bounds (it is not difficult to find that $\frac{2^{n}\cdot 2n}{(n-1)(n+3)}$ can be integer only if $n$ is $5$ or $9$). In the case $\lambda=1$, the bounds turn to the bounds in [7] and are attained by $t$-times-shortened $1$-perfect codes, $t=0,1,2,3$.

Problem 3**.**

Find more examples of $\lambda$-fold $1$-packings attaining the bound in Theorem 4. In particular, does there exist a $3$-fold $1$-packing of length $8$ and cardinality $80$, lying on bound (a)?

VI $1$-perfect unitrades and extended $1$-perfect unitrades

In this section, we consider the so-called $1$-perfect unitrades, which are a special case of two-fold $1$-packing. The term “unitrade” was introduced in [44], where latin bitrades (see Remark 4) were studied and some of their properties were described in terms of unitrades. Here we consider another kind of unitrades; however, as one can see from the sequence of remarks in this section, the class of $1$-perfect unitrades is also connected with latin bitrades (see e.g. [16], [44]) and trades of combinatorial designs (see e.g. [26]), well known in the corresponding areas of combinatorics.

A $1$-perfect unitrade is a set $T$ of vertices of $H(n,q)$ such that $|B\cap T|\in\{0,2\}$ for every radius-$1$ ball $B$ in $H(n,q)$. In the case $q=2$, appending the parity check bit turns a $1$-perfect unitrade in $H(n,2)$ to a so-called extended $1$-perfect unitrade in $H(n+1,2)$, defined as follows. A set $T$ of even-weight (or odd-weight) vertices of $H(m,2)$ such that $|B\cap T|\in\{0,2\}$ for every radius-$1$ ball $B$ in $H(m,2)$ centered in an odd-weight (respectively, even-weight) word is called an extended $1$-perfect unitrade. Clearly, removing the last coordinate in all words of an extended $1$-perfect unitrade in $H(n+1,2)$ turns it to a $1$-perfect unitrade in $H(n,2)$; so, the $1$-perfect unitrades are in one-to-one correspondence with the even-weight (or odd-weight) extended $1$-perfect unitrades.

Proposition 4**.**

(i) If $q$ is even, then non-empty $1$-perfect unitrades in $H(n,q)$ can only exist if $n$ is odd. (ii) Nonempty extended $1$-perfect unitrades in $H(n,2)$ can only exist if $n$ is even.

Proof:

We claim that if $x$ is a word from a $1$-perfect unitrade $U$ in $H(n,q)$, then $U$ has exactly one word at distance $1$ from $x$ and exactly $(n-1)(q-1)/2$ words at distance $2$ from $x$. Indeed, at first, by the unitrade definition, a radius-$1$ ball centered in $x$ has exactly one another word in $U$, say $y$. At second, each of $(n-1)(q-1)$ neighbors of $x$ that are not neighbors of $y$ has another neighbor in $U$, say $z$, and $z$ is at distance $2$ from $x$. On the other hand, each vertex at distance $2$ from $x$ has exactly $2$ common neighbors with $x$; hence, the number of such vertices in $U$ is $(n-1)(q-1)/2$. Since this value is integer, (i) is proven; (ii) is straightforward from (i). ∎

Example*.*

Consider the following set of binary words:

[TABLE]

(see Fig. 1). It is a $1$-perfect unitrade in $H(5,2)$. For example, the word $00000\not\in C$ has no neighbors in $C$, the word $00001\not\in C$ has two neighbors $00101$ and $01001$ in $C$, the word $00111\in C$ has one neighbor $00101$ in $C$ (so, the radius-$1$ ball centered in $00111$ totally has two elements of $C$). Moreover, $C$ is a $2$-fold $1$-packing attaining the bound of Theorem 4(b). Appending the first $5$ words of $C$ by $1$ and the last $5$, by [math], we get an extended $1$-perfect unitrade.

Remark 2*.*

Alternatively, the extended $1$-perfect unitrades can be defined in terms of cliques of the halved $n$-cube. A maximum collection of mutually adjacent vertices of a graph is called a maximum clique. If $n\geq 5$, then every maximum clique of $\frac{1}{2}\mathrm{H}(n,2)$ consists of $n$ vertices at Hamming distance one from some odd-weight binary word of length $n$ (such a word, of course, is not itself a vertex of $\frac{1}{2}\mathrm{H}(n,2)$). So, a set $T$ of vertices of $\frac{1}{2}\mathrm{H}(n,2)$ is an extended $1$-perfect unitrade if and only if $|B\cap T|\in\{0,2\}$ for every maximum clique $B$ in $\frac{1}{2}\mathrm{H}(n,2)$.

A (extended) $1$-perfect unitrade is called primary if it cannot be partitioned into two proper subsets that are (extended) $1$-perfect unitrades too. A (extended) $1$-perfect trade is called bipartite if it can be partitioned into two codes with minimum distance larger than $2$.

The next three remarks show connections of $1$-perfect unitrades with other areas of discrete mathematics such that design theory and latin squares. The concepts introduced there are not used in the rest of the paper.

Remark 3*.*

The $1$-perfect unitrades are related with the $1$-perfect bitrades and $1$-perfect codes in the following manner. A $1$-perfect bitrade is a pair $(T_{+},T_{-})$ of disjoint sets of vertices of $H(n,q)$ such that $|B\cap T_{+}|=|B\cap T_{-}|\in\{0,1\}$ for every radius-$1$ ball $B$. An $r$-perfect code is a set $C$ of vertices of $H(n,q)$ such that $|B\cap C|=1$ for every radius-$r$ ball $B$. If $(T_{+},T_{-})$ is a $1$-perfect bitrade, then $T_{+}\cup T_{-}$ is a $1$-perfect unitrade (but only bipartite $1$-perfect unitrades are representable in such a way). If $C$ and $C^{\prime}$ are $1$-perfect codes, then $(C\backslash C^{\prime},C^{\prime}\backslash C)$ is a $1$-perfect bitrade (but not every $1$-perfect bitrade is representable in such a way). In a similar manner, extended $1$-perfect unitrades are related to the extended $1$-perfect bitrades and the extended $1$-perfect binary codes.

Remark 4*.*

Latin unitrades (latin bitrades, latin hypercubes) are defined similarly to the extended $1$-perfect unitrades, via the intersections with the maximum cliques, but in the Hamming graph $H(n,q)$, instead of the halved $n$-cube $\frac{1}{2}\mathrm{H}(n,2)$. The latin hypercubes (in the case $n=3$, the latin squares) are known in coding theory as the unrestricted (not necessarily linear or additive) distance-$2$ MDS codes. There are constructions of binary $1$-perfect codes from latin hypercubes [41], [46], which can be adopted to constructions of $1$-perfect bitrades from latin bitrades [33] and, similarly, of $1$-perfect unitrades from latin unitrades.

Remark 5*.*

Steiner $(n,k,k-1)$ unitrades (Steiner $(n,k,k-1)$ (bi)trades, Steiner systems $S(n,k,k-1)$) can be defined similarly to the extended $1$-perfect unitrades, but the unitrade itself consists of words of weight $k$ only, while the balls $B$ in the definition are centered in the words of weight $k-1$. It is not difficult to establish that a set of binary $n$-words is a Steiner $(n,n/2,n/2-1)$ unitrade if and only if it is a constant-weight extended $1$-perfect unitrade of length $n$ (a similar relation takes place for bitrades).

The bipartiteness is a very strong property of a unitrade (actually bipartite unitrades of all kinds considered above are studied as bitrades, in the terminology popular in the theory of latin squares and latin hypercubes, or trades, in an alternative terminology popular in the design theory). A bipartite $1$-perfect unitrade (as well as latin trades and Steiner trades) is a set of nonzeros of some $\{0,\pm 1\}$-valued eigenfunction of the Hamming graph (in the case of Steiner trades, the Johnson graph) see e.g. [33] for details. Using algebraico-combinatorial tools, several distance invariant properties are derived for eigenfunctions of the Hamming graphs, see e.g. [30], [32], [51]. Below we discuss two important corollaries of these properties, trying to generalize them to the class of unitrades. We will see that, while unitrades in general cannot be represented by eigenfunctions, in oppose to bipartite unitrades, some properties, in a weaker form, can be generalized using combinatorial approaches.

VI-A The antipodality and the radius of the set

The following property for $1$-perfect codes is well-known [49], while for bipartite $1$-perfect unitrades, it is a straightforward generalization, see e.g. [33, Cor. 1].

Proposition 5**.**

Every bipartite (extended) $1$-perfect unitrade $U$ in $H(n,2)$ is antipodal: if $x\in U$, then $x+\bar{1}\in U$, where $\bar{1}$ is the all-one word and $+$ is the coordinatewise addition modulo $2$.

In other words, the inner radius $\min_{x\in U}\max_{y\in U}d(x,y)$ of a nonempty bipartite (extended) $1$-perfect unitrade $U$ is $n$. For non-bipartite case, we can only prove that the radius is larger than $n/2$.

Theorem 7**.**

Let $U$ be a $1$-perfect unitrade or an extended $1$-perfect unitrade in $H(n,2)$, let $i$ be some coordinate, $i\in\{0,\ldots,n-1\}$, and let $v$ be a binary word of length $n$.

(i) In $U$, the number of words having [math] in the $i$th coordinate equals the number of words having $1$ in it (in other words, the list of words from $U$ is an orthogonal array of strength $1$.)

(ii) An average Hamming distance from $v$ to the vertices of $U$ is $n/2$.

Proof:

Let us match any two words of $U$ differing in exactly two coordinates including $i$ or differing in only the $i$th coordinate. As follows from the definition of (extended) unitrade, every vertex $u$ from $U$ is matched to exactly one other vertex $v$ from $U$ (indeed, the radius-$1$ ball whose center differs from $u$ in only the $i$th coordinate contains $u$ and exactly one other element of $U$). Since every such $u$ and $v$ differ in the $i$th position, (i) is proven. (ii) is a simple corollary of (i). ∎

So, the inner radius $\min_{x\in U}\max_{y\in U}d(x,y)$ of a (extended) $1$-perfect unitrades is between $n/2$ and $n$. The minimum of this value remains unknown.

VI-B The minimum cardinality of a unitrade

The following fact was firstly proved in [50]. Formally, the claim was stated (in different terminology) only for the unitrades that are symmetric difference of two binary $1$-perfect codes, but the proof is applicable to any bipartite $1$-perfect unitrade.

Proposition 6**.**

The minimum cardinality of a bipartite $1$-perfect unitrade in $H(n,2)$, $n$ odd, is $2^{(n+1)/2}$.

In the modern literature, starting from [21], different variants of Propositions 6 are usually explained utilizing the distance invariant properties of the bipartite $1$-perfect unitrades, which cannot be generalized to the unrestricted case.

To prove the same fact for the non-bipartite unitrade, we adopt the combinatorial approach from [50]. It can be applied without any changes in the unrestricted case to evaluate the number of words of weight less than $n/2$ in a $1$-perfect unitrade. However, to evaluate the number of unitrade words of weight larger than $n/2$, the antipodality is utilized in [50]. The non-bipartite unitrades have no antipodality in general, so we need another argument, which occurs to be the most complicate part of the proof.

Proposition 7**.**

The minimum cardinality of a nonempty extended $1$-perfect unitrade in $H(n,2)$, $n$ even, is $2^{n/2}$. The minimum cardinality of a nonempty $1$-perfect unitrade in $H(n,2)$, $n$ odd, is $2^{(n+1)/2}$.

Proof:

Let $U$ be an extended $1$-perfect unitrade in $H(n,2)$. Assume without loss of generality that $U$ contains the all-zero word. Denote by $W$ the number of words in $U$ and by $W_{i}$ the number of words of weight $i$ in $U$. Denote by $W_{i}^{+}$ ($W_{i}^{*}$, $W_{i}^{-}$) the number of pairs $(u,v)$ of words from $U$ at distance $2$ from each other such that $\mathrm{wt}(u)=i$ and $\mathrm{wt}(v)=i+2$ ($\mathrm{wt}(v)=i$, $\mathrm{wt}(v)=i-2$, respectively). In particular, we have $W_{0}^{-}=W_{0}^{*}=W_{n}^{*}=W_{n}^{+}=0$. From the definition of a unitrade, it can be seen that

[TABLE]

but we will prove a more detailed form of (19), splitting it into two identities (20) and (21) as follows. Every vertex $u$ of weight $i$ belongs to $i$ radius-$1$ balls with the center of weight $i-1$. Each such ball has another vertex $v$ in $U$, with $\mathrm{wt}(v)=i-2$ or $\mathrm{wt}(v)=i$. The vertices $u$ and $v$ belong to exactly $2$ common radius-$1$ balls, but in the first case those two balls both have centers of weight $i-1$, while in the second case only one of the centers is of weight $i-1$. We deduce that

[TABLE]

By similar arguments,

[TABLE]

From the last two equations, we have

[TABLE]

[TABLE]

Also, trivially,

[TABLE]

We formally consider (22)–(23) as a system of $n-1$ linear equations with respect to the $n-1$ variables $W_{2}^{-}$, $W_{4}^{-}$, …, $W_{n}^{-}$, $W_{2}^{+}$, $W_{4}^{+}$, …, $W_{n-2}^{+}$, where $W_{0}^{+}$ and $W_{i}^{*}$, $i=2,4,...,n-2$, are right-side parameters. We find the solution of the system for each basis set of parameter values (temporarily, we forget about the meaning of the parameters and assume that they can possess any values, in particular, $W_{0}^{+}$ can be [math]), also calculating the value of $W$, $W=W(W_{0}^{+},W_{2}^{*},\ldots,W_{n-2}^{*})=\frac{2}{n}(W_{0}^{+}+W_{2}^{+}+\ldots+W_{n-2}^{+}+W_{2}^{*}+\ldots+W_{n-2}^{*}+W_{2}^{-}+\ldots+W_{n}^{-})$, for each of those solutions. The solutions are easy to check by substituting the values to (22) and (23).

(i) If $W_{0}^{+}=n/2$ and $W_{i}^{*}=0$, $i=2,4,...,n$, then

[TABLE]

(ii) If $W_{0}^{+}=0$, $W_{k}^{*}=1$ for some $k\in\{2,4,...,n-2\}$, and $W_{i}^{*}=0$ for every $i\in\{2,4,...,n-2\}\backslash\{k\}$, then

[TABLE]

We claim that $W>0$ for every $k$. If $k\leq n/2$, then trivially we get $\beta\geq 0$ and $W>0$. In the case $k>n/2$, we have $\beta<0$, and the inequality $W>0$ is equivalent to $nW/\beta<0$, that is,

[TABLE]

After substituting $n=2m$ and $k=2m-2l$, (24) turns to

[TABLE]

After adding $(m-l)\binom{m}{l}$ to the both sides and then dividing by $m$, we get

[TABLE]

The last inequality was proved in [43, p.50]; for completeness, we repeat the proof here.

• •

We have $m>2l$. Hence, for any $i\geq 0$ we have $m-l+i>l$, and

[TABLE]

Since $0<\frac{l}{m-l}<1$, we can find the sum of the infinite geometric series

[TABLE]

Utilizing (26) and (27), we get

[TABLE]

which validates (25) and hence (24).

So, we see that $W$ grows with the growth of any parameter $W_{i}^{*}$, $i\in\{2,4,\ldots,n-2\}$.

Since $W_{0}=1$, we have $W_{0}^{+}=n/2$. Hence, the cardinality $W$ of $U$ equals $2^{n/2}$ if $W_{i}^{*}=0$ for all $i$, and it is larger than $2^{n/2}$ if $W_{i}^{*}$ is positive for some $i$ from $\{2,4,...,n-2\}$. This proves the lower bound on the cardinality of an extended $1$-perfect unitrade. An example attending this bound is $\{(\bar{x},\bar{x}):\ \bar{x}\in\{0,1\}^{n/2}\}$. The second claim of the proposition is straightforward from the first one. ∎

VI-C Two constructions

In this subsection, we present two constructions of extended $1$-perfect unitrades. The first one (Proposition 8) is recursive and gives unitrades called reducible. The second construction (Proposition 9) gives one example of an irreducible non-bipartite extended $1$-perfect unitrade in every $H(n,2)$, $n=6,8,10,\ldots$. It happens that with these two constructions (Propositions 8 and 9), one can construct all extended $1$-perfect unitrades in $H(6,2)$ and all non-bipartite extended $1$-perfect unitrades in $H(8,2)$.

Proposition 8** (the concatenation of unitrades).**

If $U$ and $V$ are extended $1$-perfect unitrades in $H(m,2)$ and $H(n,2)$, respectively, then

[TABLE]

is an extended $1$-perfect unitrade in $H(m+n,2)$. Moreover, $W$ is a bipartite extended $1$-perfect unitrade if and only if both $U$ and $V$ are bipartite extended $1$-perfect unitrades.

Proof:

Straightforward from the definitions. ∎

We say that an extended $1$-perfect unitrade is reducible (irreducible) if it can (cannot) be represented as a concatenation (28), up to permutation of the coordinates.

Proposition 9** **(example

of an irreducible non-bipartite extended $1$-perfect unitrade).

Let $n$ be an even number larger than $4$. Define

[TABLE]

where $\langle\ldots\rangle$ denoted the linear span of the binary words understood as vectors over the binary field $\mathrm{GF}(2)$. For each $i=0,2,\ldots,{n-2}$, let $L_{i}$ consist of all words of $L$ with the values of $i$th, $(i+1)$th, $(i+2)$th, and $(i+3)$th coordinates changed to [math], $1$, $1$, [math], respectively (the coordinates are calculated modulo $n$). Then the set

[TABLE]

is an irreducible non-bipartite extended $1$-perfect unitrade. Moreover, $L_{*}+0101...01$ is constant-weight; i.e., it is also a Steiner $(n,n/2,n/2-1)$ unitrade (see Remark 5).

Proof:

For a binary word $\bar{x}=(x_{0},\ldots,x_{n-1})$, we consider the pairs $(x_{i},x_{i+1})$ of the values of two subsequent coordinates, starting from a coordinate with an even number $i=0,2,\ldots,n-2$. Such a pair is called zero if $x_{i}=x_{i+1}=0$, and non-zero otherwise; odd if $x_{i}+x_{i+1}=1$, and even otherwise.

Let us check that $L_{*}$ satisfies the definition of unitrade. The set $L$ consists of all the words with only even pairs and even number of non-zero pairs. Every word $\bar{y}$ at Hamming distance $1$ from $L$ has exactly one odd pair $(y_{i},y_{i+1})$. Clearly, $\bar{y}$ is adjacent to only one word of $L$, with $(0,0)$ or $(1,1)$ in the corresponding pair of coordinates, depending on the parity of the number of non-zero pairs in $\bar{y}$. If $(y_{i},y_{i+1})=(0,1)$, then $\bar{y}$ is adjacent to one word from $L_{i}$, with $1$ and [math] in the $(i+2)$th and $(i+3)$th coordinates, respectively. If $(y_{i},y_{i+1})=(1,0)$, then $\bar{y}$ is adjacent to one word from $L_{i-2}$, with [math] and $1$ in the $(i-2)$th and $(i-1)$th coordinates. It is easy to see that the two words above, one from $L$ and one from $L_{i}$ or $L_{i-2}$, are the only words from $L_{*}$ adjacent to $\bar{y}$.

The other group of words at Hamming distance $1$ from $L_{*}$ consists of words $\bar{y}$ with exactly three odd pairs. Moreover, two of these pairs are in four consequent coordinates with numbers $i$, ${i+1}$, ${i+2}$, ${i+3}$ and have the values $(0,1)$, $(1,0)$, respectively. These two pairs are determined uniquely. Changing the values in the third odd pair to $(0,0)$ or $(1,1)$, we obtain the two words of $L_{*}$ (more tightly, of $L_{i}$) at Hamming distance $1$ from $\bar{y}$.

So, a sphere of radius $1$ centered in a word of odd weight contains [math] or $2$ elements of $L_{*}$; i.e., $L_{*}$ is an extended $1$-perfect unitrade by the definition.

Since the complement of any word from $L_{i}$ is not in $L_{*}$, we see from Proposition 5 that $L_{*}$ cannot be bipartite.

To ensure that $L_{*}$ is irreducible, we consider a graph on the $n$ coordinates as the vertices, two coordinate being adjacent if $L_{*}$ contains two words differing only in these coordinates. For every $i$ from $\{0,2,\ldots,{n-2}\}$, there are two words in $L$ and $L_{i}$, respectively, that differ in the $i$th and $(i+2)$th coordinates (with the values $1,1,0,0$ and $0,1,1,0$ in the consequent four coordinates starting from the $i$th coordinate); similarly, there are two words in $L$ and $L_{i}$, respectively, that differ in the $i$th and $(i+3)$th coordinates (with the values $1,1,1,1$ and $0,1,1,0$). It follows that the considered graph is connected, which obviously not the case for a reducible unitrade.

The last statement of the proposition is obvious from the following three facts: (i) for every $\bar{x}=(x_{0},\ldots,x_{n-1})\in L+0101...01$ and every $j$ from $\{0,2,\ldots,{n-2}\}$, we have $\{x_{j},x_{j+1}\}=\{0,1\}$; (ii) we have the same for every $i$ from $\{0,2,\ldots,{n-2}\}$, every $\bar{x}$ from $L_{i}+0101...01$, and every $j$ from $\{0,2,\ldots,{n-2}\}\backslash\{i,i+2\}$; (iii) for every $i$ from $\{0,2,\ldots,{n-2}\}$ and every $\bar{x}$ from $L_{i}+0101...01$, we have $(x_{i},x_{i+1},x_{i+2},x_{i+3})=(0,0,1,1)$. ∎

VI-D Classification of small unitrades

The bipartite extended $1$-perfect unitrades in $H(n,2)$, $n=6,8,10$, where classified in [33] in terms of bitrades (see Remark 3 for the definition). Here, we briefly describe the results of the classification of all small unitrades up to length $10$, mainly focusing on the non-bipartite case. The classification algorithm is described in the Appendix.

The only non-bipartite unitrade of length $6$, up to equivalence, is $L_{*}(6)$. The two nonequivalent non-bipartite unitrades of length $8$ are $L_{*}(8)$ and $L_{*}(6)10\cup L_{*}(6)01$. There are $38$ equivalence classes of primary unitrades of length $10$ [28], with unitrade cardinalities 32, 40, 48, 48, 50, 56, 56, 56, 58, 62, 62, 64, 64, 64, 70, 70, 70, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 76, 80, 80, 80, 80, 86, 88, 88, 96, 96, 96; eight of them (marked by bold) are bipartite; eleven (underlined) have constant-weight representatives. The unique non-bipartite antipodal unitrade of length $10$ is notable; it is a vertex orbit of cardinality $80$ under the automorphism group of the $(10,40,4)$ Best code (see [18]). There are only $4$ nonempty non-primary unitrades of length $10$, of cardinality 32+32, 32+32, 40+40, and 40+40 (in particular, we have convinced that all $2$-fold $1$-packings of size $96$ in $H(9,2)$ are primary unitrades).

Classification of

$\lfloor n/2\rfloor$-regular triangle-free subgraphs of the halved $n$-cube

As follows from Corollary 2, any binary even-weight $\lambda$-fold $1$-packing attaining a bound in Theorem 5 induces an $\lfloor n(\lambda-1)/2\rfloor$-regular subgraph of the halved $n$-cube. Moreover, any subgraph corresponds to a $2$-fold $1$-packing if and only if it has no induced triangles. As a result, optimal binary even-weight $2$-fold $1$-packings of length up to $10$ can be classified with the classification of $\lfloor n/2\rfloor$-regular triangle-free subgraphs of $\frac{1}{2}\mathrm{H}(n,2)$, $n\leq 10$. The following pseudocode describes a breadth-first algorithm for the search all inequivalent subsets of vertices of the halved $n$-cube that induce a connected $\lfloor n/2\rfloor$-regular triangle-free subgraph. The partial solutions found at each step are validated using the double-counting approach [27, §10.2] based on the orbit-stabilizer theorem (in practice, the isomorph rejection and double-counting validation were processed only at the first three recursive steps for partial solutions, and for the final solutions).

define RECURSION($s$): # $s$ is the step number

if $T_{s-1}^{+}=T_{s}^{+}=\{\}$:  

    FOUND_SOLUTION() # *record the solution $(T_{0},T_{1})$* 

else if $T_{s-1}^{+}=\{\}$: 

    if $T$ is new, up to equivalence: # *isomorph rejection* 

        RECURSION($s+1$) # *go to the next step* 

else:   

    choose $v$ from $T_{s-1}^{+}$ 

    $T_{s-1}^{+}:=T_{s-1}^{+}\backslash\{v\}$ 

    for all $\lfloor n/2\rfloor$-subsets $N$ of the neighborhood of $v$ 

      such that $N\cup T$ is triangle-free do: 

        $N^{+}:=N\backslash T$ # *new vertices to add* 

        $T_{s}^{+}:=T_{s}^{+}\cup N^{+}$ 

        $T:=T\cup N^{+}$

        RECURSION($s$)  

        $T_{s}^{+}:=T_{s}^{+}\backslash N^{+}$ 

        $T:=T\backslash N^{+}$ 

    $T_{s-1}^{+}:=T_{s-1}^{+}\cup\{v\}$ 

$T:=\{00...0\}$ # start from the all-zero word

$T_{0}^{+}:=T$ # $T_{i}^{+}$ keep the chosen vertices with the “unsolved” neighborhood

$T_{i}^{+}:=\{\},\quad i=1,2,\ldots$

RECURSION(1)

After the classification of connected subgraphs, they can be combined in all inequivalent ways to form a disconnected subgraphs (for $n\leq 10$, this step is straightforward, and the number of connected components is at most $2$).

For even $n$, the algorithm above finds all extended $1$-perfect unitrades. In [33], the “bipartite” modification of this algorithm was used to classify the extended $1$-perfect bitrades (essentially, bipartite unitrades). Without the bipartiteness, the number of search branches becomes essentially larger, but still doable for the length up to $10$. The classification took almost ten hours of computer time.

Acknowledgements

The authors are grateful to the referees for the detailed reviews, which enabled us to greatly improve the presentation of the results.

The research was carried out at the Sobolev Institute of Mathematics at the expense of the Russian Science Foundation, Grants 14-11-00555 (Sections IV-A, V-A,  VI-CD) and 18-11-00136 (Sections II, III,  IV-B, V-B, and VI-AB).