On the Maximum Size of Block Codes Subject to a Distance Criterion

TL;DR

This paper derives a general formula for the maximum size of block codes with a given minimum distance, applicable to various alphabets and distance measures, and extends to zero-error capacity and asymptotic bounds.

Contribution

It introduces a universal formula for maximum code size based on the distance distribution, generalizing the Gilbert-Varshamov bound and applicable to broad code and distance types.

Findings

Formula characterizes maximum code size via distance distribution.

Recovers and generalizes Gilbert-Varshamov bound.

Provides bounds on asymptotic code rate.

Abstract

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a codeword. We demonstrate that the number of such balls that cover the entire code alphabet cannot exceed this maximum size. Our approach can be applied to codes $i)$ with elements over arbitrary code alphabets, and $ii)$ under a broad class of distance measures, thereby ensuring the generality of our formula. Our formula indicates that the maximum code size can be fully characterized by the cumulative distribution function of the distance measure evaluated at two independent and identically distributed random codewords. When the two random codewords assume a uniform distribution over the entire code alphabet, our formula recovers and obtains a natural generalization of the Gilbert-Varshamov (GV) lower bound. We also establish a general formula for the zero-error capacity of any sequence of channels. Finally, we extend our study to the asymptotic setting, where we establish first- and second-order bounds on the asymptotic code rate subject to a normalized minimum distance constraint.

Equations

$$M_{n}^{\ast}(d)=\frac{1}{\inf_{P_{X^{n}}}\Pr\big{[}\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}<d\big{]}},$$

$$\mu (u^n,v^n)={\mu }_{\min }\triangleq \underset{{\hat{x}}^n,x^n\in {\mathcal{X}}^n}{\min }\mu ({\hat{x}}^n,x^n)\text{if}{u}^n=v^n.$$

$$\underset{{\hat{x}}^n,x^n\in \mathcal{C}\text{ and }{\hat{x}}^n\ne x^n}{\min }\mu ({\hat{x}}^n,x^n)\ge d.$$

$$M_n^{\ast }(d)\triangleq \max \left\{M\in \mathbb{N}:\exists (n,M,d)\text{mbox}-code\right\},$$

$$\displaystyle M_{n}^{\ast}(d)=\frac{1}{\inf_{P_{X^{n}}}\Pr\big{[}\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}<d\big{]}}.$$

$$\Pr\big{[}\tilde{\mu}({\hat{X}}^{n},X^{n})<d\big{]}\geq\frac{1}{M_{n}^{\ast}(d)},$$

$$\inf_{P_{X^{n}}}\Pr\big{[}\tilde{\mu}({\hat{X}}^{n},X^{n})<d\big{]}\geq\frac{1}{M_{n}^{\ast}(d)}.$$

$$a_1\triangleq \underset{x^n\in {\mathcal{X}}^n}{\inf }\mathrm{Pr}[X^n\in \mathcal{B}(x^n)],$$

$$a_2\triangleq \underset{x^n\in {\mathcal{X}}^n\setminus \mathcal{B}(u_1^n)}{\inf }\mathrm{Pr}[X^n\in \mathcal{B}(x^n)\setminus \mathcal{B}(u_1^n)].$$

$$a_i\triangleq \underset{x^n\in {\mathcal{X}}^n\setminus {\cup }_{j=1}^{i-1}\mathcal{B}(u_j^n)}{\inf }\mathrm{Pr}[X^n\in \mathcal{B}(x^n)\setminus {\cup }_{j=1}^{i-1}\mathcal{B}(u_j^n)]$$

$$\mathrm{Pr}[\tilde{\mu }({\hat{X}}^n,X^n)<d]$$

$$=$$

$$\ge$$

$$=$$

$$>$$

$$=$$

$$\ge$$

$$\underset{x^n\in {\mathcal{D}}_j}{\inf }{\int }_{\mathcal{B}(x^n)}\mathrm{d}{P}_{X^n}({\hat{x}}^n)$$

$$\ge$$

$$\displaystyle\Pr[\tilde{\mu}({\hat{X}}^{n\ast},X^{n\ast})<d]=\sum_{x^{n}\in{\cal C}^{\ast}}\big{[}P_{X^{n\ast}}(x^{n})\big{]}^{2}=\sum_{x^{n}\in{\cal C}^{\ast}}\frac{1}{|{\cal C}^{\ast}|^{2}}=\frac{1}{M_{n}^{\ast}(d)},$$

$$\min_{P_{X^{n}}}\Pr\big{[}\tilde{\mu}({\hat{X}}^{n},X^{n})<d\big{]}=\frac{1}{M_{n}^{\ast}(d)}.$$

$$\mathrm{Pr}[\tilde{\mu }({\hat{X}}^{n\circ },X^{n\circ })<d]=\frac{1}{|\mathcal{S}|},$$

$$\inf_{P_{X^{n}}}\Pr\big{[}\tilde{\mu}({\hat{X}}^{n},X^{n})<d\big{]}=0.$$

$$C_0\triangleq \underset{n\ge 1}{\sup }\frac{1}{n}\log {\Omega }_n.$$

$$C_0$$

$$\mu ({\hat{x}}^n,x^n)$$

$$\mathcal{U}({\hat{x}}^n)\triangleq \bigcap _{x^n\in {\mathcal{C}}_n:x^n\ne {\hat{x}}^n}{\mathcal{D}}_{{\hat{x}}^n}(x^n)$$

$$C_0$$

$$=$$

$$C_0\ge -\frac{1}{n}\log \underset{P_X}{\inf }\mathrm{Pr}\left[\sum _{y\in \mathcal{Y}}{P}_{Y|X}(y|\hat{X})P_{Y|X}(y|X)>0\right].$$

$$\mu ({\hat{x}}^n,x^n)$$

$$=$$

$$C_0$$

Full text

I Introduction

Given a code alphabet ${\cal X}$, the determination of the maximal size $M_{n}^{\ast}(d)$ of a block code ${\cal C}\subseteq{\cal X}^{n}$ with pairwise minimum distance no less than $d$ and block length $n<\infty$ has been a long-standing problem in information and coding theory. One can use $M^{\ast}_{n}(d)$ to obtain an upper bound of the expurgated error exponent [1] and also to characterize the capacity of a graph [2]. Some well-known bounds on $M^{\ast}_{n}(d)$ include the linear programming upper bound [3] and Gilbert-Varshamov (GV) lower bound [3, 4, 5]. Other famous upper bounds include the Singleton, Plotkin, and Elias bounds [6]. However, these bounds are not tight in general. Since finite-length bounds are usually difficult to obtain, researchers have focused on asymptotic analyses in which blocklength $n$ tends to infinity. One then considers the limit of the code rate $(1/n)\log M_{n}^{\ast}(d)$ subject to a normalized distance constraint $d/n\geq\delta$. Many asymptotic bounds have been derived; see, for example [3, 7, 8, 9, 10, 4, 11, 12, 13] and the references therein.

A natural question then beckons. Can one derive a “meta-result” concerning the maximum code size subject to a fixed minimum distance $M^{\ast}_{n}(d)$ that recovers some of the above-mentioned bounds as special cases? In [14], using a graph-theoretic framework, Motzkin and Straus derived such a result which implies an exact formula for $M^{\ast}_{n}(d)$ under the condition that ${\cal X}$ is finite. See also a related result by Korn [15]. It is then natural to ask if there exists an analogous result for more general code alphabets, e.g., uncountable alphabets. This is precisely the purpose of this paper. Our contributions are as follows:

We propose an iterative construction of a set of balls, each centered at a codeword and of a fixed radius $d$. We then show that the number of such balls that cover the entire code alphabet cannot exceed the maximum code size subject to a fixed minimum distance $d$. Consequently, we prove that $M_{n}^{\ast}(d)$ for an arbitrary code alphabet can be completely determined by the minimum probability (over all distributions) that two i.i.d. random vectors ${\hat{X}}^{n}$ and $X^{n}$ are at distance less than $d$ from each other, i.e.,

[TABLE]

where $\mu(\cdot,\cdot)$ is the (possibly asymmetric) distance measure. Our formula can be used to recover and obtain a natural generalization of the GV bound. 2. 2.

Based on our result, a general formula for the zero-error capacity [16] of any sequence of channels can be obtained. 3. 3.

Finally, we derive a general expression for the maximum asymptotic code rate $\frac{1}{n}\log M_{n}^{\ast}(n\delta)$ as $n$ approaches infinity such that the relative minimum distance is at least $\delta$. Our proof also characterizes a second-order bound (cf. Lemma 2) on the maximum asymptotic code rate.

The term $\Pr[\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}<d]$ can be regarded as the information spectrum [17, 18] (or cumulative distribution function) of the distance measure $\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}$. Posing the problem of finding $M^{\ast}_{n}(d)$ as an optimization problem over a continuous variable (the distribution $P_{X^{n}}$) can potential yield new approaches for finding codes with improved performances over existing codes.

The rest of the paper is organized as follows. The exact formula for $M_{n}^{\ast}(d)$ is presented in Section II. This formula is then used to obtain a general formula for the zero-error capacity of a sequence of channels. A family of lower bounds to $M^{\ast}_{n}(d)$ is presented in Section III; also included here is the demonstration that the finite length GV lower bound can be recovered from our formula. Extensions to the asymptotic regime are studied in Section IV. Finally, open problems are discussed in Section V.

II Maximal Code Size Attainable under a Minimum Pairwise Distance

We first introduce the notation used in this paper. An $(n,M)$-code over alphabet ${\cal X}^{n}$ denotes a set of $M$ codewords, each of which belongs to ${\cal X}^{n}$ [19]. A distance measure $\mu(\cdot,\cdot)$ is a real-valued function with domain ${\cal X}^{n}\times{\cal X}^{n}$ which satisfies

[TABLE]

Here, we do not require $\mu(\cdot,\cdot)$ to be symmetric or satisfy the triangle inequality but can be arbitrary as long as it admits its minimum from a point to itself.

An $(n,M,d)$-code ${\cal C}$ denotes an $(n,M)$-code with the minimum pairwise distance among codewords at least $d$, i.e.,

[TABLE]

The maximal code size $M_{n}^{\ast}(d)$ subject to a pairwise minimum distance $d$ is given by

[TABLE]

where $\mathbb{N}$ is the set of positive integers. For convenience, a code that satisfies (3) is referred to as a distance-$d$ code. Throughout this paper, ${\hat{X}}^{n}$ and $X^{n}$ denote two independent random variables with a common distribution $P_{X^{n}}$ over ${\cal X}^{n}$.

II-A Distance Spectrum Formula of $M^{\ast}_{n}(d)$

We now present a general formula for the maximum size $M^{\ast}_{n}(d)$ of distance-$d$ codes over an arbitrary code alphabet ${\cal X}$ (not necessarily countable) and general distance measure $\mu(\cdot,\cdot)$.

Theorem 1

Fix an arbitrary code alphabet ${\cal X}$ and a distance measure $\mu(\cdot,\cdot)$ that satisfies (2). For all $n\geq 1$ and $d>\mu_{\min}$, we have

[TABLE]

Proof:

We first prove the validity of (5) under the assumption that $M_{n}^{\ast}(d)<\infty$. Its extension to $M_{n}^{\ast}(d)=\infty$ will be done next.

Subject to the condition that $M_{n}^{\ast}(d)$ is finite, the equality in (5) can be proved in two steps. We first show that for every distribution $P_{X^{n}}$ over ${\cal X}^{n}$, the following inequality holds:

[TABLE]

where for convenience, we denote $\tilde{\mu}({\hat{x}}^{n},x^{n})\triangleq\min\{\mu({\hat{x}}^{n},x^{n}),\mu(x^{n},{\hat{x}}^{n})\}$; hence,

[TABLE]

The proof is then completed by exhibiting a distribution $P_{X^{n\ast}}$ that results in equality in (7); consequently, given that $M_{n}^{\ast}(d)$ is finite, the infimum in (5) can be replaced by a minimum.

Achievability (Validation of (6) under finite $M_{n}^{\ast}(d)$): Fix a distribution $P_{X^{n}}$ over ${\cal X}^{n}$ and an arbitrarily small $\epsilon>0$. Let

[TABLE]

where ${\cal B}(x^{n})\triangleq\{{\hat{x}}^{n}\in{\cal X}^{n}:\tilde{\mu}({\hat{x}}^{n},x^{n})<d\}$. Find an element $u_{1}^{n}$ in ${\cal X}^{n}$ such that $p_{1}\triangleq\Pr[X^{n}\in{\cal B}(u_{1}^{n})]<a_{1}+\epsilon$. Note that the existence of $u_{1}^{n}$ is guaranteed by the definition of the infimum. Let

[TABLE]

Find an element $u_{2}^{n}$ in ${\cal X}^{n}\setminus{\cal B}(u_{1}^{n})$ such that $p_{2}\triangleq\Pr[X^{n}\in{\cal B}(u_{2}^{n})\setminus{\cal B}(u_{1}^{n})]<a_{2}+\epsilon$. We repeat this procedure to obtain

[TABLE]

and an $u_{i}^{n}$ in ${\cal X}^{n}\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})$ with $p_{i}\triangleq\Pr[X^{n}\in{\cal B}(u_{i}^{n})\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})]<a_{i}+\epsilon$ for $i=3,4,\ldots,k$ until $\cup_{j=1}^{k}{\cal B}(u_{j}^{n})$ covers the entire ${\cal X}^{n}$, i.e., ${\cal X}^{n}\setminus\cup_{j=1}^{k}{\cal B}(u_{j}^{n})=\emptyset$ but ${\cal X}^{n}\setminus\cup_{j=1}^{k-1}{\cal B}(u_{j}^{n})\not=\emptyset$. Two observations are made: $i)$ $\{u_{1}^{n},u_{2}^{n},\ldots,u_{k}^{n}\}$ is a distance-$d$ code and hence by the definition of $M_{n}^{\ast}(d)$ and its assumed finiteness, $k\leq M_{n}^{\ast}(d)$ is a finite integer so the above procedure is repeated at most $M_{n}^{\ast}(d)$ times; $ii)$ $\sum_{j=1}^{k}p_{j}=1$. Denoting ${\cal D}_{i}\triangleq{\cal B}(u_{i}^{n})\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})$ and noting ${\cal D}_{i}\cap{\cal D}_{j}=\emptyset$ for $i\neq j$ and ${\cal X}^{n}=\cup_{j=1}^{k}{\cal D}_{j}$ (i.e., $\{{\cal D}_{j}\}_{j=1}^{k}$ is a partition of ${\cal X}^{n}$), we can derive the following chain of inequalities:

[TABLE]

where ${\bf 1}\{\cdot\}$ is the set indicator function; (14) holds because

[TABLE]

(15) follows from the definition of $p_{j}$; (16) holds since $p_{j}<a_{j}+\epsilon$; (17) applies since $\sum_{j=1}^{k}p_{j}=1$; (18) is a consequence of the Cauchy-Schwarz inequality;111The Cauchy-Schwarz inequality can be used to assert that 1=\big{(}\sum_{j=1}^{k}1\cdot p_{j}\big{)}^{2}\leq\big{(}\sum_{j=1}^{k}1^{2}\big{)}\big{(}\sum_{j=1}^{k}p_{j}^{2}\big{)}=k\sum_{j=1}^{k}p_{j}^{2}. and the last inequality in (19) follows from $k\leq M_{n}^{\ast}(d)$. The proof of (6) is completed by noting that the above derivations hold for arbitrarily small $\epsilon$. 2. 2.

Converse (Equality of (7) under finite $M_{n}^{\ast}(d)$): Let $P_{X^{n\ast}}$ be the uniform distribution over a distance-$d$ code ${\cal C}^{\ast}$ that achieves $M_{n}^{\ast}(d)$. We then have

[TABLE]

where $|{\cal C}^{\ast}|$ denotes the cardinality of ${\cal C}^{\ast}$.

The above two steps complete the proof of

[TABLE]

subject to finite $M_{n}^{\ast}(d)$.

When $M_{n}^{\ast}(d)=\infty$, again, let ${\cal C}^{\ast}$ denote an infinite distance-$d$ code that achieves $M_{n}^{\ast}(d)$. Then, any finite subset ${\cal S}$ of ${\cal C}^{\ast}$ is a distance-$d$ code. Using a derivation similar to that leading to (23) gives that

[TABLE]

where $P_{X^{n\circ}}$ is the uniform distribution over ${\cal S}$. As $|{\cal S}|$ can be made arbitrarily large,

[TABLE]

This completes the proof. ∎

Some remarks concerning Theorem 1 are in order. First, the theorem can be applied to an arbitrary code alphabet and any distance measure satisfying (2). Its generality thus extends the study of the maximal code size of distance-$d$ codes from the conventional finite code alphabets and the Hamming distance to, for example, ${\cal X}=[0,1)$ and the Euclidean distance (cf. Example 1).

Secondly, the crux of the proof of Theorem 1 is the observation that the entire space ${\cal X}^{n}$ can be covered by $k$ “open” balls of radius $d$ with $k\leq M_{n}^{\ast}(d)$,222Here “open” means a strict inequality is used to define the ball. where the radius is defined via the distance $\tilde{\mu}(\cdot,\cdot)$. In addition, the selection of the center $u_{i}^{n}$ of the next ball ${\cal B}(u_{i}^{n})$ is chosen such that $p_{i}=\Pr[X^{n}\in{\cal B}(u_{i}^{n})\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})]$ is $\epsilon$-close to its minimum possible value and therefore $k$ can be made as large as possible, ideally as close to $M_{n}^{\ast}(d)$ as possible.

Thirdly, as noted by Korn [15], when the code alphabet ${\cal X}^{n}$ is finite, the optimization problem $\inf_{P_{X^{n}}}\Pr[\tilde{\mu}({\hat{X}}^{n},X^{n})<d]$ corresponds exactly to the minimization of the quadratic form ${\bf p}\mathbb{A}{\bf p}^{\tt T}$, where ${\bf p}$ is the row vector formed by listing the probability masses of $P_{X^{n}}$ and $\mathbb{A}$ is the corresponding $|{\cal X}|^{n}\times|{\cal X}|^{n}$ matrix with entries given by ${\bf 1}\{\tilde{\mu}({\hat{x}}^{n},x^{n})<d\}$. This quadratic optimization problem was considered by Korn [15] in his study of the maximization of Gallager’s lower bound for the zero-error capacity of discrete memoryless channels (DMCs) [20]. The same solution can also be found in Motzkin and Strass’ work [14], where the order of the maximal complete graph contained in a finite graph is considered. Here, instead of iteratively removing one codeword from any two codewords within distance $d$ until the size of the set of candidate codewords is reduced to $M_{n}^{\ast}(d)$ as suggested by Korn’s technique in [15], we define a “proper” notion of progress to iteratively add codewords to a distance-$d$ code. Specifically, we select a representative vector $u_{i}^{n}$ in some “$\epsilon$-neighborhood” defined as $\{u^{n}\in{\cal X}^{n}:\Pr[X^{n}\in{\cal B}(u^{n})\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})]<\inf_{x^{n}\in{\cal X}^{n}}\Pr[X^{n}\in{\cal B}(x^{n})\setminus\cup_{j=1}^{i-1}{\cal B}(u_{j}^{n})]+\epsilon\}$ for a given distribution $P_{X^{n}}$. This selection is repeated until the entire code alphabet can be covered by the union of radius-$d$ balls centered at $u_{i}$’s. The assumed finiteness of $M_{n}^{\ast}(d)$ ensures that the iterative selection will terminate. Note that the proof of Theorem 1 is not restricted to code alphabets that are finite (cf. [14, 15]). In addition to being applicable to general arbitrary code alphabets, it provides a different perspective of the general formula in (5).

II-B Relation of Theorem 1 to the Zero-Error Capacity

We now show that Theorem 1 can be used to establish a general formula for the zero-error capacity [16] for arbitrary channels. This result complements the general formula for the (vanishing error) capacity of arbitrary channels considered by Verdú and Han in [21].

Definition 1** (Zero-error capacity)**

Let $\Omega_{n}$ be the maximum code size that can be transmitted error-free (i.e., with exactly zero error probability) over the channel $P_{Y^{n}|X^{n}}$. Then, the zero-error capacity for a sequence of channels $\{P_{Y^{n}|X^{n}}\}_{n=1}^{\infty}$ is defined as

[TABLE]

Theorem 2** (General zero-error capacity)**

The zero-error capacity for an arbitrary sequence of channels $\{P_{Y^{n}|X^{n}}\}_{n=1}^{\infty}$ (not necessarily with countable alphabets) can be expressed as

[TABLE]

where

[TABLE]

and $E(x^{n};{\cal T})\triangleq\Pr[Y^{n}\in{\cal T}|X^{n}=x^{n}]$ for every measurable ${\cal T}$.

Note that when ${\cal Y}$ is finite, (29) implies that $\mu({\hat{x}}^{n},x^{n})=0$ if and only if there exists an $y^{n}$ such that the channel $P_{Y^{n}|X^{n}}$ maps both ${\hat{x}}^{n}$ and $x^{n}$ to $y^{n}$ with positive probabilities.

Proof:

For ${\hat{x}}^{n}$ and $x^{n}$ with $\mu({\hat{x}}^{n},x^{n})=1$, we denote by ${\cal D}_{{\hat{x}}^{n}}(x^{n})$ and ${\cal D}_{x^{n}}({\hat{x}}^{n})$ two subsets of ${\cal Y}^{n}$ satisfying $E({\hat{x}}^{n};{\cal D}_{{\hat{x}}^{n}}(x^{n}))=E(x^{n};{\cal D}_{x^{n}}({\hat{x}}^{n}))=1$ and ${\cal D}_{{\hat{x}}^{n}}(x^{n})\cap{\cal D}_{x^{n}}({\hat{x}}^{n})=\emptyset$. Note that the existence of disjoint ${\cal D}_{{\hat{x}}^{n}}(x^{n})$ and ${\cal D}_{x^{n}}({\hat{x}}^{n})$ is guaranteed by ${\cal D}_{{\hat{x}}^{n}}(x^{n})\subset{\cal T}$ and ${\cal D}_{x^{n}}({\hat{x}}^{n})\subset{\cal Y}^{n}\setminus{\cal T}$, where ${\cal T}$ is the set to make $\mu({\hat{x}}^{n},x^{n})=1$ in (29).

Then, the next two steps verify that $\Omega_{n}$ in Definition 1 is equal to $M_{n}^{\ast}(1)$.

Achievability: To show that a code ${\cal C}_{n}$ that achieves $M_{n}^{\ast}(1)$ can be decoded error-free, we define

[TABLE]

and note that for every distinct pair of ${\hat{x}}^{n}$ and $x^{n}$ in the code, we have ${\cal U}({\hat{x}}^{n})\cap{\cal U}(x^{n})=\emptyset$ because ${\cal D}_{{\hat{x}}^{n}}(x^{n})\cap{\cal D}_{x^{n}}({\hat{x}}^{n})=\emptyset$. As a result, the decoder that decodes $y^{n}$ to $x^{n}\in{\cal C}_{n}$ if $y^{n}\in{\cal U}(x^{n})$, and to an arbitrary codeword if $y^{n}\not\in\cup_{x^{n}\in{\cal C}_{n}}{\cal U}(x^{n})$, has zero error probability. Hence, $\Omega_{n}\geq M_{n}^{\ast}(1)$. 2. 2.

Converse: Suppose $\Omega_{n}>M_{n}^{\ast}(1)$. Let the code that achieves $\Omega_{n}$ be denoted by ${\cal C}_{n}$. Then, there exist a partition of disjoint subsets $\{{\cal U}(x^{n})\}_{x^{n}\in{\cal C}_{n}}$ on ${\cal Y}^{n}$ such that the decoder that decodes $y^{n}$ to $x^{n}$ when $y^{n}\in{\cal U}(x^{n})$ has zero error probability. Therefore, $E({\hat{x}}^{n};{\cal U}(x^{n}))=0$ whenever ${\hat{x}}^{n}$ and $x^{n}$ are two distinct elements in ${\cal C}_{n}$. This implies that $E({\hat{x}}^{n};{\cal U}({\hat{x}}^{n}))=1$ since $\sum_{x^{n}\in{\cal C}_{n}}E({\hat{x}}^{n};{\cal U}(x^{n}))=1$. As a result of $E({\hat{x}}^{n};{\cal U}({\hat{x}}^{n}))=E(x^{n};{\cal Y}^{n}\setminus{\cal U}({\hat{x}}^{n}))=1$, we have $\mu(\hat{x}^{n},x^{n})=1$ for all distinct pairs of ${\hat{x}}^{n}$ and $x^{n}$ in ${\cal C}_{n}$, implying $M_{n}^{\ast}(1)\geq\Omega_{n}$. Thus, a contradiction to the assumption that $M_{n}^{\ast}(1)<\Omega_{n}$ is obtained.

By noting that $\mu({\hat{x}}^{n},x^{n})=\mu(x^{n},{\hat{x}}^{n})$ and $\Omega_{n}=M_{n}^{\ast}(1)$, the proof of the theorem is completed by noting that

[TABLE]

∎

The next corollary presents an alternative derivation of Korn’s lower bound to the zero-error capacity for DMCs [15].

Corollary 1

For DMCs with finite channel input alphabet ${\cal X}$ and finite channel output alphabet ${\cal Y}$,

[TABLE]

Proof:

We first note that for DMCs with finite ${\cal X}$ and ${\cal Y}$, (29) can be equivalently written as

[TABLE]

where ${\cal T}^{c}={\cal Y}^{n}\setminus{\cal T}$. We thus derive from Theorem 2 that

[TABLE]

This concludes the proof of Corollary 1. ∎

We remark that the inequality in (40) may be strict for DMCs. An example in which (40) is a strict inequality is given in [16, Fig. 2], for which the five channel input letters form the vertices of a pentagon graph and an edge exists whenever $\mu(\hat{x},x)=0$ as defined in (29). The zero-error capacities of the pentagon and some special generalizations were later established by Lovász in [2]. However, for most other DMCs, the determination of the zero-error capacity remains a major open problem in information theory and combinatorics.

III Implications of the Distance Spectrum Formula for $M_{n}^{\ast}(d)$

In this section, further explorations based on the theoretical result in the previous section are conducted. Section III-A shows that the GV lower bound for discrete alphabets can be recovered from an information spectrum perspective. Section III-B provides examples that leverage specifically uniform codeword distributions. In particular, two of the examples considers continuous code alphabets.

III-A Relation to the Gilbert-Varshamov bound

An immediate consequence of Theorem 1 is that a family of lower bounds to $M_{n}^{\ast}(d)$ can be obtained by evaluating $L_{X^{n}}(d)\triangleq 1/\Pr[\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}<d]$ for different distributions $P_{X^{n}}$. This implies even if we do not use an optimal distribution $P_{X^{n}}$, we may still be able to obtain good lower bounds to the optimal code size.

From this perspective, the Gilbert-Varshamov (GV) lower bound [4] can be recovered with a uniform distribution over all possible codewords. Specifically, consider a finite code alphabet ${\cal X}$ with $|{\cal X}|=Q$ and the Hamming distance measure $\mu(\cdot,\cdot)$. Let the components of $X^{n}=(X_{1}\ X_{2}\ \ldots\ X_{n})$ be i.i.d. and uniform over ${\cal X}$. This choice yields

[TABLE]

Hence,

[TABLE]

which is exactly the Gilbert-Varshamov (GV) lower bound [4]. The same observation has been stated by Kolesnik and Krachkovsky in [22, pp. 1446].

III-B Uniform Distribution (UD) Lower Bounds

The converse proof of Theorem 1 shows that $M_{n}^{\ast}(d)$ can actually be achieved using a distribution which is uniform over an appropriate subset of ${\cal X}^{n}$ (that is, over an optimal code). The discussion in the previous subsection also suggests that considering uniform $X^{n}$ over (a proper subset of) ${\cal X}^{n}$ may result in good lower bounds. Thus,

[TABLE]

based on uniform $P_{X^{n}}$ may form an important family of lower bounds to $M_{n}^{\ast}(d)$. For convenience, we will refer to this family of bounds as uniform distribution (UD) lower bounds.

Example 1

Here we derive an UD lower bound to $M_{2}^{\ast}(d)$ for Euclidean distance $\mu(\cdot,\cdot)$ and a bounded code alphabet ${\cal X}=[0,1)$. Taking $P_{X^{2}}$ to be the uniform distribution over ${\cal X}^{2}$ and letting $Z_{i}\triangleq({\hat{X}}_{i}-X_{i})^{2}$ yields that for $d>0$,

[TABLE]

where f_{Z}(z)=\big{(}\frac{1}{\sqrt{z}}-1\big{)}{\bf 1}\{0\leq z<1\}. We can check that

[TABLE]

Via a procedure suggested by the proof of Theorem 1, we can actually obtain

[TABLE]

This indicates that there is room for improving the UD lower bound in (47) in this situation and the codeword selection procedure in the proof of Theorem 1 could be further explored for finding a better lower bound.

Example 2

We continue from the previous example but turn to a modified rectilinear distance measure $\mu({\hat{x}}^{n},x^{n})\triangleq\sum_{i=1}^{n}\min\{1-|{\hat{x}}_{i}-x_{i}|,|{\hat{x}}_{i}-x_{i}|\}$. We again derive UD lower bounds to $M_{2}^{\ast}(d)$ for ${\cal X}=[0,1)$. Taking $P_{X^{2}}$ to be the uniform distribution over ${\cal X}^{2}$ and noting that each of $\mu({\hat{X}}_{1},X_{1})$ and $\mu({\hat{X}}_{2},X_{2})$ is uniform over $[0,1/2)$ and they are independent of each other, we have

[TABLE]

which implies

[TABLE]

To improve the above lower bound for $0\leq d\leq\frac{1}{2}$, we adopt another distribution $P_{X^{2\circ}}$. Let $X_{1}^{\circ}$ and $X_{2}^{\circ}$ be uniformly distributed over $\{0,d,2d,\ldots,(\ell-1)d\}$ and be independent of each other, where $\ell\triangleq\lfloor 1/d\rfloor$. Then, $\Pr[\mu({\hat{X}}_{1}^{\circ},X_{1})+\mu({\hat{X}}_{2}^{\circ},X_{2})<d]=\Pr[{\hat{X}}_{1}^{\circ}=X_{1}]\Pr[{\hat{X}}_{2}^{\circ}=X_{2}]=\frac{1}{\ell^{2}}$, and M_{2}^{\ast}(d)\geq L_{X^{2\circ}}(d)=\big{(}\lfloor 1/d\rfloor\big{)}^{2}. Thus, when $1/d$ is an even integer, the lower bound in (50) is doubled by taking a discrete distribution.

We can further improve $L_{X^{2\circ}}(d)$ for certain values of $d$ by introducing dependency among components. For $d=\frac{1}{k}$ with $k\geq 2$, let $X_{1}^{\ast}$ be uniform over $\{0,\frac{d}{2},d,\ldots,(2k-1)\frac{d}{2}\}$ and let $X_{2}^{\ast}$ be uniform over $\{X_{1}^{\ast}$, $(X_{1}^{\ast}+d)\,\text{mod}\,1$, $(X_{1}^{\ast}+2d)\,\text{mod}\,1$, $\ldots$, $(X_{1}^{\ast}+(k-1)d)\,\text{mod}\,1\}$. Thus, $X^{2\ast}$ is uniform over $2k^{2}$ points, each of which is at least at distance $1/k$ from all others. This implies $\Pr[\mu({\hat{X}}^{2\ast},X^{2\ast})<\frac{1}{k}]=\frac{1}{2k^{2}}$, and $M_{2}^{\ast}(\frac{1}{k})\geq L_{X^{2\ast}}(\frac{1}{k})=2k^{2}$, which improves $L_{X^{2\circ}}(1/k)=k^{2}$ for all integer $k\geq 2$. By a similar technique to the Hamming bound, we can obtain

[TABLE]

*where $|\cdot|$ here denotes the area in a 2-dimensional real plane. Therefore, $L_{X^{2\ast}}(1/k)=M_{2}^{\ast}(1/k)$ for $k\geq 2$ is indeed tight. *

For $\frac{1}{2}\leq d<1$, we introduce $P_{X^{2\diamond}}$ with strongly dependent components as $X_{2}^{\diamond}=X_{1}^{\diamond}$ with probability one. Then, noting that $\mu({\hat{X}}_{2}^{\diamond},X_{2}^{\diamond})$ is equal to $\mu({\hat{X}}_{1}^{\diamond},X_{1}^{\diamond})$ with probability one, we have $\Pr[\mu({\hat{X}}^{2\diamond},X^{2\diamond})<d]=\Pr[\mu({\hat{X}}_{1}^{\diamond},X_{1}^{\diamond})+\mu({\hat{X}}_{2}^{\diamond},X_{2}^{\diamond})<d]=\Pr[\mu({\hat{X}}_{1}^{\diamond},X_{1}^{\diamond})<\frac{d}{2}]$. With $X_{1}^{\diamond}$ uniform over $\{0,\frac{d}{2},\ldots,(j-1)\frac{d}{2}\}$, where $j\triangleq\lfloor 2/d\rfloor$, we obtain $\Pr[\mu({\hat{X}}_{1}^{\diamond},X_{1}^{\diamond})<\frac{d}{2}]=\Pr[{\hat{X}}_{1}^{\diamond}=X_{1}^{\diamond}]=1/j$, which implies $M_{2}^{\ast}(d)\geq L_{X^{2\diamond}}(d)=\lfloor 2/d\rfloor$. Accordingly, when $\frac{1}{2}<d\leq\frac{2}{3}$, $L_{X^{2\diamond}}(d)=3>\lceil L_{X^{2}}(d)\rceil=2$, and hence improvement can be obtained by introducing dependence among components.

Example 3

In this example, we demonstrate a case that $M_{n}^{\ast}(d)$ can be exactly determined and hence its relation with the UD lower bound $L_{X^{n}}(d)$ can be quantitatively examined.

Let the distance measure be given by

[TABLE]

where ${\hat{x}}^{n}$ and $x^{n}$ are in $\{0,1\}^{n}$, and $\kappa_{n}(x^{n})\triangleq x_{n}2^{n-1}+x_{n-1}2^{n-2}+...+x_{2}2^{1}+x_{1}$ is the binary representation of $x^{n}=(x_{1}\ x_{2}\ \ldots\ x_{n})$. In other words, $\mu({\hat{x}}^{n},x^{n})$ is the absolute difference between two decimal numbers $\kappa_{n}({\hat{x}}^{n})$ and $\kappa_{n}(x^{n})$, and is a separable distance measure [23, Def. 1].

Since $\kappa_{n}(x^{n})$ is an integer in $\{0,1,2,\ldots,2^{n}-1\}$, it can be easily seen that for $d>0$,

[TABLE]

where $\lceil\cdot\rceil$ is the ceiling function. Notably, one of the uniform $X^{n}$’s that results in $L_{X^{n}}(d)=M_{n}^{\ast}(d)$ has support $\{0,\lceil d\rceil,2\lceil d\rceil,\ldots,(M_{n}^{\ast}(d)-1)\lceil d\rceil\}$, and there are exactly $\lceil d\rceil$ optimizers that can achieve $M_{n}^{\ast}(d)$.

We then recall that (44) has illustrated that $G_{n}(d)$ can be regarded as a special case of the UD lower bound with uniform $X^{n}$ over the entire ${\cal X}^{n}$. As such, we derive

[TABLE]

showing that $G_{n}(d)$ is strictly less than $M_{n}^{\ast}(d)$ except when $\lceil d\rceil=1$ and $\lceil d\rceil\geq 2^{n}$. This result confirms that the finite length GV lower bound is not tight in general.

We close this example by noting that an upper bound $U_{n}(d)$ for $M_{n}^{\ast}(d)$ can also be provided based on Theorem 1. If there exists $U_{n}(d)$ such that $U_{n}(d)\geq 1/\Pr{[\min\{\mu({\hat{X}}^{n},X^{n}),\mu(X^{n},{\hat{X}}^{n})\}<d}]$ for all $P_{X^{n}}$’s, then

[TABLE]

Now setting $j=j(n,d)\triangleq 2^{n}/\lceil d\rceil$, we derive

[TABLE]

where (60) and (61) hold because ${\hat{X}}^{n}$ and $X^{n}$ are i.i.d., and (62) again follows from the Cauchy-Schwarz inequality. This immediately gives

[TABLE]

which is precisely $M_{n}^{\ast}(d)$. $\Box$

IV Extensions to the Asymptotic Regime

We now extend the result in Theorems 1 to the asymptotic regime in which the length $n$ of the code goes to infinity. In what follows, $\log$ denotes the natural logarithm. A distance spectrum formula for the largest code rate $R=\log(M)/n$ subject to a normalized minimum distance $\delta=d/n$ can be obtained on the basis of Theorem 1 in a straightforward manner:

[TABLE]

The formula of $R^{\ast}_{n}(\delta)$ in (64) provides a quantitative characterization of the largest code rate attainable for an $(n,M,n\delta)$-code, based on which a first-order expression for the largest asymptotic code rate attainable for a sequence of $(n,M,n\delta)$-codes can be obtained when the normalized distance measure is uniformly bounded.

Theorem 3

(Largest Asymptotic Code Rate) Fix an arbitrary code alphabet ${\cal X}$ and a (sequence of) general distance measures $\mu(\cdot,\cdot)$ that satisfy the condition mentioned in Theorem 1 and also satisfy

[TABLE]

Then,

[TABLE]

where

[TABLE]

Proof:

The proof can be found in Appendix A. In particular, an upper bound on the second-order term of $R_{n}^{\ast}(\delta)$ is also provided (cf. Lemma 2). ∎

The above theorem indicates that $R_{n}^{\ast}(\delta)$ and $\sup_{P_{X^{n}}}J_{X^{n}}(\delta)$ are asymptotically close. In fact, the proof in Appendix A shows that $R_{n}^{\ast}(\delta)\geq\sup_{P_{X^{n}}}J_{X^{n}}(\delta)$ for every $n$. However, the proof of the upper bound is significantly more involved and requires delicate twisting of probability distributions [24].

A lower bound to $R_{n}^{\ast}(\delta)$ can be obtained when $P_{X^{n}}$ is i.i.d. and $\mu(\cdot,\cdot)$ is additive, e.g., the Hamming distance.

Corollary 2

Assume that ${\cal X}=\{\alpha_{1},\alpha_{2},...\alpha_{Q}\}^{n}$ and $\mu(\cdot,\cdot)$ is the Hamming distance measure. For any $\delta>0$, one has

[TABLE]

where

[TABLE]

is the binary Kullback-Leibler divergence [25].

Proof:

The proof is deferred to Appendix B. ∎

Using a large deviations technique, we can slightly improve (69) by the addition of a logarithmic term as for $Q=2$ and $0<\delta<1/2$,

[TABLE]

Although Jiang and Vardy [26, Thm. 1] have shown, by using a graph-theoretic framework, that the achievable second-order term in (71) is at least $(\log n)/n$, which is slightly stronger than the term $(\log n)/(2n)$, Eq. (71) provides some additional insight into the suboptimality of choosing ${\hat{X}}^{n}$ and $X^{n}$ with i.i.d. components (since our evaluation of the relevant distance spectrum is asymptotically tight when employing distributions to yield dependent elements of ${\hat{X}}^{n}$ and $X^{n}$).

V Conclusion and Future Work

In this paper, we developed an exact formula for the maximal size of distance-$d$ codes for arbitrary alphabets and general distance measures. The implications of the established formula were discussed. The extension to the asymptotic regime was also explored. Some natural directions for future work includes:

• •

Understanding the structure of optimal or even “good” distributions $P_{X^{n}}$ to give lower bounds on the optimal code size. For example, based on our numerical experiments, we know that the optimal distribution may not be unique. Studying the binary Hamming distance for small block lengths suggests that there may be an optimizer whose marginals are uniform on each coordinate.

• •

Seek $i)$ a similar formula of the minimum code size subject to a covering radius constraint (cf. [27]) and $ii)$ a formula of maximal code size under a minimum multi-wise distance constraint (cf. [28]). The latter would constitute a generalization of Turán’s Theorem.

Appendix A Proof of Theorem 3

The theorem can be verified via the following two lemmas. The first lemma shows that for arbitrary distance measures, $R_{n}^{\ast}(\delta)$ is lower-bounded by $\sup_{P_{X^{n}}}J_{X^{n}}(\delta)$. The second lemma proves that $R_{n}^{\ast}(\delta)$ is upper bounded by $\sup_{X^{n}}J_{X^{n}}(\delta)+\Theta(\frac{1}{\sqrt{n}})$ when the normalized distance measure is uniformly bounded. Then, the two lemmas imply Theorem 3.

Lemma 1

Fix an arbitrary code alphabet and an arbitrary distance measure that satisfies (2). Then,

[TABLE]

Proof:

It suffices to prove that for every $P_{X^{n}}$,

[TABLE]

Denote

[TABLE]

as the large deviation rate function with respect to the normalized random distance $\frac{1}{n}\mu({\hat{X}}^{n},X^{n})$, where $\varphi_{X^{n}}(\theta)\triangleq\frac{1}{n}\log{\mathbb{E}}[e^{\theta\mu({\hat{X}}^{n},X^{n})}]$. An elementary property333 The large deviation rate function $I_{X^{n}}(a)$ is convex, and admits its global minimum $\min_{a\in\Re}I_{X^{n}}(a)=0$ at $a=(1/n){\mathbb{E}}[\mu({\hat{X}}^{n},X^{n})]$ [29]. of the large deviation rate function gives that

[TABLE]

Thus, it suffices to prove (73) under the condition that $\delta<\frac{1}{n}{\mathbb{E}}[\mu({\hat{X}}^{n},X^{n})]$ (since [math] is a trivial lower bound). Let $Y\triangleq n\delta-\mu({\hat{X}}^{n},X^{n})$ and note from the previous condition that ${\mathbb{E}}[Y]<0$. We then derive from Markov’s inequality that for $\theta>0$,

[TABLE]

Applying the fact that $\left.\frac{\partial}{\partial\theta}M_{Y}(\theta)\right|_{\theta=0}=\mathbb{E}[Y]<0$ and the convexity of $M_{Y}(\theta)$ over $\theta\in\Re$, we obtain

[TABLE]

which complete the proof of (73). ∎

Lemma 2

Fix an arbitrary code alphabet and an arbitrary distance measure that satisfies both (2) and (65). Then, given that $P_{X^{n}}$ is the optimizer of $\sup_{X^{n}}J_{X^{n}}(\delta)$, we have

[TABLE]

for those $\delta$ satisfying $\sup_{P_{X^{n}}}J_{X^{n}}(\delta)>0$.

Proof:

Following the notations used in the proof of Lemma 1, we define the twisted distribution of $Y$ as

[TABLE]

Then,

[TABLE]

where $\theta^{\ast}$ is the minimizer of $\inf_{\theta\in\Re}M_{Y}(\theta)$. Note that since $P_{X^{n}}$ is the optimizer for $\sup_{P_{X^{n}}}J_{X^{n}}(\delta)$ and only those $\delta$ satisfying $\sup_{P_{X^{n}}}J_{X^{n}}(\delta)>0$ is considered, we can infer from (75) that ${\mathbb{E}}[Y]<0$ and hence $0<\theta^{\ast}<\infty$. By noting that $\Pr[Y^{(\theta)}>0]$ is positive,444 If $\Pr[Y^{(\theta^{\ast})}>0]=0$, then $-Y^{(\theta^{\ast})}$ is a non-negative random variable. We can therefore apply Markov’s inequality to obtain $\Pr[-Y^{(\theta^{\ast})}\geq\epsilon]\leq{\mathbb{E}}[-Y^{(\theta^{\ast})}]/\epsilon=0$ since the minimizer of $\inf_{\theta\in\Re}M_{Y}(\theta)$ must validate $E[Y^{(\theta^{\ast})}]=0$ [29]. As a result, $\Pr[-\epsilon<Y^{(\theta^{\ast})}\leq 0]=1$ for arbitrary $\epsilon>0$, which implies $\Pr[Y^{(\theta^{\ast})}=0]=1$. As $Y^{(\theta^{\ast})}$ and $Y$ have the same support, we conclude $\Pr[Y=0]=1$, thereby resulting a contradiction to ${\mathbb{E}}[Y]<0$. we let $W$ be a nonnegative random variable with distribution

[TABLE]

Then, (84) can be rewritten as

[TABLE]

Using the fact that ${\mathbb{E}}[Y^{(\theta^{\ast})}]=0$ [29, Thm. 9.2], we obtain555 $\Pr[Y^{(\theta^{\ast})}>0]>0$ and ${\mathbb{E}}[Y^{(\theta^{\ast})}]=0$ jointly imply ${\mathbb{E}}\{[Y^{(\theta^{\ast})}]^{2}\}>0$ and ${\mathbb{E}}\{[Y^{(\theta^{\ast})}]^{4}\}>0$, which justifies (89).

[TABLE]

where

[TABLE]

and

[TABLE]

Notably, for a bounded distance measure $\mu(\cdot,\cdot)$, $\varphi_{X^{n}}(\theta)$ is guaranteed to be fourth-order differentiable. Using Jensen’s inequality, we derive

[TABLE]

and hence

[TABLE]

We conclude from all the above derivations that

[TABLE]

which concludes the proof of (80). ∎

We complete the proof of Theorem 3 by remarking that with probability one, $(1/n)\mu({\hat{X}}^{n},X^{n})$ is not only bounded, but uniformly upper bounded in the block length $n$, and so are its moments and cumulants. Since a twisted random variable generated from $(1/n)\mu({\hat{X}}^{n},X^{n})$ must have the same support as $(1/n)\mu({\hat{X}}^{n},X^{n})$, its twisted moments as well as twist cumulants are also uniformly bounded. Accordingly, $\varphi_{X^{n}}^{(4)}(-\theta^{\ast})=O(1)$ and $\varphi_{X^{n}}^{\prime\prime}(-\theta^{\ast})=O(1)$, based on which (80) implies $R_{n}^{\ast}(\delta)\leq\sup_{X^{n}}J_{X^{n}}(\delta)+\Theta(\frac{1}{\sqrt{n}})$.

Appendix B Proof of Corollary 2

Proof:

When the distance measure $\mu(\cdot,\cdot)$ is additive and the components of $X^{n}$ are i.i.d. with generic distribution $P_{X}$, $I_{X^{n}}(a)$ as defined in (74) exhibits a single-letter expression for all block lengths $n$ as:

[TABLE]

where \varphi_{X}(\theta)\triangleq\log{\mathbb{E}}\big{[}e^{\theta\mu(\hat{X},X)}\big{]}. Thus, when $X^{n}$ has i.i.d. components, (68) is simplified to

[TABLE]

This and (72) lead to

[TABLE]

As long as ${\cal X}=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{Q}\}$ is finite and $\mu(\cdot,\cdot)$ is the Hamming distance measure, we have

[TABLE]

and

[TABLE]

where $b_{X}\triangleq 1-\sum_{i=1}^{Q}P_{X}^{2}(\alpha_{i})={\mathbb{E}}[\mu({\hat{X}},X)]$. Again, according to the property of the large deviation rate function [29], we have

[TABLE]

According to the Cauchy-Schwarz inequality, $0\leq b_{X}\leq 1-1/Q=(Q-1)/Q$. Combining (108), (111) and (112), we conclude that for $\delta>0$,

[TABLE]

∎

Acknowledgements

The work of Ling-Hua Chang and Po-Ning Chen is supported by the Ministry of Science and Technology (MoST), Taiwan, under grant 105-2221-E-009-009-MY3. The work of Carol Wang and Vincent Tan is supported by a Singapore Ministry of Education (MoE) Tier 2 grant (R-263-000-B61-112). The work of Yunghsiang S. Han is support by the National Natural Science Foundation of China (Grant No. 61671007).

A special acknowledgement is given to Dr. Mladen Kovačević, who brought [22] to the authors’ attention. The authors sincerely thank the anonymous reviewers for their constructive feedback to improve the quality of the paper.