On the VC-Dimension of Binary Codes

TL;DR

This paper explores the limits of binary codes with constrained VC-dimension and minimum distance, providing new upper and lower bounds on their asymptotic rates.

Contribution

It introduces novel upper bounds derived from existing theorems and new lower bounds using Gilbert-Varshamov type arguments for codes with VC-dimension constraints.

Findings

Two upper bounds on asymptotic code rates with VC-dimension constraints

Two lower bounds based on Gilbert-Varshamov type constructions

Insights into the trade-offs between VC-dimension, code length, and minimum distance

Abstract

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $\delta n$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.

Equations

$$R_{LP}(\delta ):=\underset{0\le u\le 1-2\delta }{\min }\{1+g(u^2)-g(u^2+2\delta u+2\delta )\}$$

$$|\mathcal{C}|\le \sum _{i=0}^{dn}\left(\genfrac{}{}{0ex}{}{n}{i}\right)$$

$$C(d,\delta )\le \frac{2d}{\delta +2d}\cdot h\left(⟨\frac{\delta +2d}{2}⟩\right).$$

$$W\le 2dn|\mathcal{C}|.$$

$$\mathbf{E}[W]\ge \frac{sn\cdot \delta n}{n-sn+1}\left(|\mathcal{C}|-\sum _{i=0}^{dn}\left(\genfrac{}{}{0ex}{}{sn}{i}\right)\right).$$

$$\left(((\delta +2d)s-2d)-\frac{2d}{n}\right)|\mathcal{C}|\le s\delta \sum _{i=0}^{dn}\left(\genfrac{}{}{0ex}{}{sn}{i}\right).$$

$$C(d,\delta )\le \underset{0\le s\le 1-2\delta }{\min }\left\{s\cdot h\left(⟨\frac{d}{s}⟩\right)+(1-s)R_{LP}\left(\frac{\delta }{1-s}\right)\right\}.$$

$$|{\cal C}|=\sum_{{\bm{u}}\in{\mathcal{C}}|_{[sn]}}\left|\mathcal{Z}({\bm{u}})\right|\leq\Big{|}{\cal C}|_{[sn]}\Big{|}\cdot\max_{{\bm{u}}\in{\mathcal{C}}|_{[sn]}}\left|\mathcal{Z}({\bm{u}})\right|.$$

$$A(n,\delta n,wn)\ge \frac{\left(\genfrac{}{}{0ex}{}{n}{wn}\right)}{{\sum }_{i=0}^{\delta n/2-1}\left(\genfrac{}{}{0ex}{}{wn}{i}\right)\left(\genfrac{}{}{0ex}{}{n-wn}{i}\right)}.$$

$$\displaystyle C(d,\delta)\geq\begin{cases}h(w)-{\displaystyle\max_{0\leq x\leq\delta/2}}\Big{[}w\,h\Big{(}\frac{x}{w}\Big{)}+(1-w)h\Big{(}\frac{x}{1-w}\Big{)}\Big{]}&\textup{if }w<\frac{1}{2}\\ 1-h(\delta)&\textup{otherwise}.\end{cases}$$

$$M\ge \frac{|S{|}^2}{4|{\mathcal{B}}_S(\delta n-1)|}$$

$${\mathcal{B}}_S(\delta n-1):=\{(\mathbf{w},{\mathbf{w}}^{\prime })\in S\times S:\text{dist}(\mathbf{w},{\mathbf{w}}^{\prime })\le \delta n-1\}.$$

$$\mathrm{Pr}\{X=u,Y=v\}=\{\begin{array}{ll}P((u,v))& \text{if }(u,v)\in E_G\\ 0& \text{otherwise.}\end{array}$$

$${\mathbf{E}}_P(f):=\sum _{e\in E_G}P(e)f(e).$$

$$P_{\gamma }(e):=\frac{1}{n}|\{i\in \{1,2,\dots ,n\}:e_i=e\}|.$$

$${\mathbf{E}}_{P_{\gamma }}(f)=\sum _{e\in E_G}{P}_{\gamma }(e)f(e)$$

$${\Gamma }_n(G;f,U):=\{\gamma \in {\Gamma }_n(G):{\mathbf{E}}_{P_{\gamma }}(f)\in U\}.$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log |{\Gamma }_n(G;f,U)|=\underset{P\in \mathcal{M}(G;f,U)}{\sup }{H}_P(Y|X).$$

$$S_n(d)=S_n([0,d]):=\{L_G(e_1)L_G(e_2)\cdots L_G(e_n):e_1{e}_2\cdots e_n\in {\Gamma }_n(G;f,[0,d])\}.$$

$$f^{(1)}(⟨e,e^{\prime }⟩)=f(e),f^{(2)}(⟨e,e^{\prime }⟩)=f(e^{\prime })$$

$$\Delta (⟨e,e^{\prime }⟩)=\{\begin{array}{ll}1& \text{if }{L}_G(e)\ne L_G(e^{\prime })\\ 0& \text{otherwise.}\end{array}$$

$$\mathcal{F}(U)$$

$$\mathcal{G}(U,\delta )$$

$$\displaystyle\begin{split}&R_{MA}(d,\delta):=\sup_{p\in[0,d]}\{2{\mathcal{F}}(p)-{\mathcal{G}}(p,\delta)\}\\ =&\sup_{p\in[0,d]}\Big{\{}2\sup_{\begin{subarray}{c}P\in{\mathcal{M}}(G):\\ {\bf E}_{P}(f)=p\end{subarray}}H_{P}(Y|X)-\sup_{\begin{subarray}{c}Q\in{\mathcal{M}}(G\times G):\\ {\bf E}_{Q}(f^{(i)})=p,\,i=1,2\\ {\bf E}_{Q}(\Delta)\in[0,\delta)\end{subarray}}H_{Q}(Y|X)\Big{\}}.\end{split}$$

$$\frac{\log M}{n}\ge R_{MA}(d,\delta )-o(1).$$

$$S_n(U_{p,\epsilon }):=\{L_G(e_1)L_G(e_2)\cdots L_G(e_n):e_1{e}_2\cdots e_n\in {\Gamma }_n(G;f,U_{p,\epsilon })\},$$

$${\mathcal{B}}_{S_n(U_{p,\epsilon })}(\delta n-1):=\{(\mathbf{w},{\mathbf{w}}^{\prime })\in S_n(U_{p,\epsilon })\times S_n(U_{p,\epsilon }):\text{dist}(\mathbf{w},{\mathbf{w}}^{\prime })\le \delta n-1\}.$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log |S_n(U_{p,\epsilon })|=\underset{n\to \infty }{\lim }\frac{1}{n}\log |{\Gamma }_n(G;f,U_{p,\epsilon })|=\mathcal{F}(U_{p,\epsilon }),$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log |{\mathcal{B}}_{S_n(U_{p,\epsilon })}(\delta n-1)|=\underset{n\to \infty }{\lim }\frac{1}{n}\log |{\Gamma }_n(G\times G;\phi ;U_{p,\epsilon }\times U_{p,\epsilon }\times [0,\delta ))|=\mathcal{G}(U_{p,\epsilon },\delta ).$$

$$\frac{\log M}{n}\ge 2\mathcal{F}(p)-\mathcal{G}(p,\delta )-o(1).$$

$$[A_{G;f}(\mathbf{x}){]}_{u,v}=\{\begin{array}{ll}{2}^{-\mathbf{x}\cdot f((u,v))}& \text{if }(u,v)\in E_G\\ 0& \text{otherwise},\end{array}$$

Full text

1 Introduction

Let ${\mathcal{C}}\subseteq\{0,1\}^{n}$ be a binary code of length $n$ and rate $R=\tfrac{1}{n}\log_{2}|{\mathcal{C}}|$. In this paper, we study the relation between the rate of the code and two fundamental properties: its minimum (Hamming) distance and its Vapnik–Chervonenkis (VC) dimension [19]. Recall that the Hamming distance between two codewords is the number of positions in which they differ; the minimum distance of ${\mathcal{C}}$, which is the smallest Hamming distance between any pair of codewords, plays an important role in coding theory. Recall that the projection of ${\mathcal{C}}$ onto a coordinate set $I\subseteq[n]:=\{1,2,\ldots,n\}$, denoted ${\mathcal{C}}|_{I}$, is the set of all possible values assigned to these coordinates by the codewords in ${\mathcal{C}}$. The code ${\mathcal{C}}$ is said to shatter $I$ if ${\mathcal{C}}|_{I}=\{0,1\}^{|I|}$. The VC-dimension of ${\mathcal{C}}$, which is the maximum size of a coordinate set that is shattered by ${\mathcal{C}}$, plays an important role in statistical learning theory and computational geometry [1, 6, 9].

Our goal in this paper is to analyze codes of simultaneously large minimum distance and small VC-dimension. Loosely speaking, we note that fixing a rate and striving to optimize one of these properties is expected to essentially be the worst possible for the other property. Indeed, on the one hand, it is well known that random linear codes achieve the Gilbert-Varshamov bound [7, 20], which is the best known lower bound on the rate of binary codes under a minimum distance constraint, yet clearly their VC-dimension is the largest possible (attained by any information set). On the other hand, by the Sauer–Shelah lemma [17, 18], the VC-dimension at any given rate is essentially minimized by any Hamming ball of a suitable radius, yet clearly the minimum distance of a Hamming ball is equal to $1$, the smallest possible. These extremal observations demonstrate the tension between increasing the minimum distance and decreasing the VC-dimension.

Besides being an interesting combinatorial problem, finding codes that have a large minimum distance as well as a small VC-dimension also admits the following coding-theoretic motivation. Suppose that a binary code ${\mathcal{C}}$ with minimum distance $\Delta$ and VC-dimension $D$ is used over an errors and erasures channel. Suppose there were $e$ erasures, and we are now interested in detecting whether any errors have fallen in the remaining $n-e$ coordinates. Let $t_{e}\in\{0,1,\ldots,n-e\}$ be the maximal number of errors that the code can guarantee to detect, and let $\pi_{e}\in\{0,1,\ldots,2^{n-e}\}$ the maximal number of distinct error sequences (of length $n-e$) that the code can guarantee to detect. The error detection threshold pertaining to each of these quantities is the maximal number of erasures $e$ such that the respective quantity is nonzero. If $e<\Delta-1$, then the minimum distance of the projection of ${\mathcal{C}}$ onto the remaining $n-e$ coordinates is at least $\Delta-e>1$. Hence, the code can correct at least $\lfloor(\Delta-e)/2\rfloor$ errors and thus in this case $t_{e}>0$. Similarly, if $e<n-D$ then ${\mathcal{C}}$ cannot shatter the remaining $n-e\,(>D)$ coordinates. Thus, there must be error sequences that result in vectors that are not contained in the projection of ${\mathcal{C}}$ onto the remaining $n-e$ coordinates; such error sequences can clearly be detected, hence $\pi_{e}>0$. Adopting this viewpoint, it is interesting to seek codes for which both error detection thresholds are high, namely codes with a large minimum distance and a small VC-dimension. We are interested in the maximum size of such codes.

In what follows, we consider the asymptotic formulation of the problem. For any333For $d\geq 1/2$ it is easy to see that the rate $R$ is always equal to $1$, which is not interesting. Therefore we limit $d$ in the interval $[0,1/2]$. $d,\delta\in[0,\frac{1}{2}]$, we say that a rate $R$ is $(d,\delta)$-achievable if for any $N$ there exists a binary code ${\mathcal{C}}$ of length $n\geq N$, rate at least $R$, VC-dimension at most $\lfloor dn\rfloor$, and minimum distance at least $\lceil\delta n\rceil$. We are interested in characterizing $C(d,\delta)$, which we define to be the supremum of all $(d,\delta)$-achievable rates. For brevity, we assume throughout that $dn$ and $\delta n$ are integers, as this does not affect the asymptotic behavior.

In Section 2 we derive two upper bounds for $C(d,\delta)$. The first is obtained as a simple asymptotic corollary of a result by Haussler [8], and the second is derived via a shortening approach that combines the Sauer–Shelah lemma [17, 18] (controlling the VC-dimension) and the linear programming bound [15] (controlling the minimum distance). In Section 3 we present two lower bounds for $C(d,\delta)$. Both these bounds are obtained via GV-type arguments (controlling the minimum distance) applied to constant-weight and Markov-type sets respectively (whose structure controls the VC-dimension).

2 Upper Bounds

We first briefly review upper bounds on $C(d,\delta)$ that can be easily deduced from known results. To begin, one can clearly ignore either the minimal distance constraint or the VC-dimension constraint.

When accounting only for the minimal distance constraint, the best known upper bound is the second MRRW bound given by McEliece, Rodemich, Rumsey, and Welch [15] as follows:

[TABLE]

with $g(x):=h((1-\sqrt{1-x})/2)$. Here and throughout this paper we define $h(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ to be the binary entropy function. The following is direct.

Lemma 1**.**

$C(d,\delta)\leq R_{LP}(\delta).$**

When accounting only for the VC-dimension constraint, the size of a code ${\cal C}$ with VC-dimension $dn$ can be upper bounded by the Sauer–Shelah lemma [17, 18]

[TABLE]

and so the following is evident.

Lemma 2**.**

$C(d,\delta)\leq h(d).$**

In [8] Haussler directly addressed the problem of bounding the size of codes with restricted minimal distance and VC-dimension. In his setting, the VC-dimension is a bounded constant. However, from the results there the following bound on $C(d,\delta)$ can still be deduced. For a number $a\geq 0$ we define $\left\langle a\right\rangle:=\min(a,\frac{1}{2})$. For a code ${\mathcal{C}}$ we define the *unit distance graph * UD(${\mathcal{C}}$) whose vertex set is all codewords in ${\mathcal{C}}$ and two codewords ${\bm{x}},{\bm{y}}$ are adjacent if their Hamming distance $\textup{dist}({\bm{x}},{\bm{y}})=1$.

Lemma 3** (Corollary to [8, Theorem 1]).**

[TABLE]

Proof.

Let ${\mathcal{C}}$ be a length-$n$ binary code with VC-dimension at most $dn$ and minimum distance $\delta n$. Suppose $0\leq s\leq 1$. We choose a random subset $I\subseteq[n]:=\{1,2,\dots,n\}$ of size $sn$ uniformly. For each codeword ${\bm{u}}\in{\mathcal{C}}|_{I}$, we define its weight $w({\bm{u}})$ as the number of codewords in ${\mathcal{C}}$ such that its projection on $I$ is equal to ${\bm{u}}$. Let $E$ be the edge set of the unit distance graph UD(${\mathcal{C}}|_{I}$), and define the weight of an edge $e=\{{\bm{u}},{\bm{v}}\}$ as $w(e)=\min\{w({\bm{u}}),w({\bm{v}})\}$. Put $W=\sum_{e\in E}w(e)$, and note that $W$ is a random variable depending on the random choice of $I$. The bound follows by estimating ${\bf E}[W]$, the expectation of $W$, in two ways. First, we claim that for any $I\subset[n]$,

[TABLE]

On the other hand, we can bound ${\bf E}[W]$ from below:

[TABLE]

(Please refer to [14, Lemma 5.14] for the proof of (2) and (3).) Thus we have

[TABLE]

For any $s>\frac{2d}{\delta+2d}$ and sufficient large $n$, we can get $|{\mathcal{C}}|=O(\sum_{i=0}^{dn}{sn\choose i})$, and hence $C(d,\delta)\leq s\cdot h(\langle{d}/{s}\rangle).$ The result follows directly. ∎

We shall next combine Lemma 1 and Lemma 2 to obtain an improved upper bound. Throughout this paper, we define $0/0=0$.

Theorem 1**.**

[TABLE]

Proof.

Let ${\mathcal{C}}$ be a length-$n$ binary code with VC-dimension at most $dn$ and minimum distance $\delta n$. Choose $s\in[0,1-2\delta]$, and consider the projection of ${\cal C}$ on $[sn]=\{1,2,\dots,sn\}$. Of course the VC-dimension of ${\mathcal{C}}|_{[sn]}$ is also at most $dn$, and so its rate can be bounded by Lemma 2. For any given prefix ${\bm{u}}\in{\mathcal{C}}|_{[sn]}$, we denote the set of its possible suffixes by $\mathcal{Z}({\bm{u}})\subset\{0,1\}^{(1-s)n}$, i.e., for any ${\bm{v}}\in{\mathcal{Z}}({\bm{u}})$ there exists a codeword ${\bm{x}}\in{\mathcal{C}}$ such that ${\bm{x}}$ is the concatenation of ${\bm{u}}$ and ${\bm{v}}$. Clearly, $\mathcal{Z}({\bm{u}})$ is a code of length $(1-s)n$ and minimal distance $\delta n$, and so its rate can be bounded by the second MRRW bound. Then our result follows from

[TABLE]

∎

3 Lower Bounds

A general procedure to obtain lower bounds on $C(d,\delta)$ is the following.

1. (i)

Pick some subset $S$ of the Hamming cube $\{0,1\}^{n}$ that has some “nice” structure. 2. (ii)

Compute a generalized GV bound for subset $S$, namely a lower bound on the size of the largest code of minimum distance at least $\delta n$ where all codewords belong to $S$. 3. (iii)

Find an upper bound for the VC-dimension $dn$ of any subset of $S$ that has minimum distance at least $\delta n$. 4. (iv)

Combine the bounds (ii)-(iii).

In the following two subsections, we will show two ways to choose “nice” subsets of the Hamming cube and calculate the corresponding bounds.

3.1 Constant Weight Codes

Here we choose subset $S$ to be the collection of all codewords with some constant weight.

Lemma 4**.**

*Suppose $\delta\in[0,\frac{1}{2}]$ and $w\in[0,1]$. Let ${\mathcal{C}}$ be a binary code of length $n$, constant weight $wn$, and minimum distance $\delta n$. Then the VC-dimension of ${\mathcal{C}}$ is at most $(w-\delta/{2})n+1$. *

Proof.

Suppose the VC-dimension of ${\mathcal{C}}$ is $dn$. Without loss of generality, we assume that the first $dn$ coordinates are shattered. Then there exist two codewords ${\bm{x}}=x_{1}x_{2}\cdots x_{n}$ and ${\bm{y}}=y_{1}y_{2}\cdots y_{n}$ such that $x_{i}=1$ for $1\leq i\leq dn$ and $y_{i}=1$ for $1\leq i\leq dn-1$ and $y_{dn}=0$. Hence $|\textup{supp}({\bm{x}})\cap\textup{supp}({\bm{y}})|\geq dn-1$. On the other hand, $\textup{dist}({\bm{x}},{\bm{y}})=2wn-2|\textup{supp}({\bm{x}})\cap\textup{supp}({\bm{y}})|$, which is at least $\delta n$. Therefore $\delta n\leq 2wn-2|\textup{supp}({\bm{x}})\cap\textup{supp}({\bm{y}})|\leq 2wn-2(dn-1)$. This proves the result. ∎

Let $A(n,\delta n,wn)$ denote the maximum size of length-$n$ binary code with constant weight $wn$ and minimum distance $\delta n$. The following GV-type bound is well-known.

Lemma 5**.**

[TABLE]

Now we are ready to state our first lower bound for $C(d,\delta)$.

Theorem 2**.**

Let $d,\delta\in[0,\frac{1}{2}]$, and let $w=d+\frac{\delta}{2}$. Then

[TABLE]

Proof.

If $w<\frac{1}{2}$, plug it into (4) and take the asymptotic form, then the result follows directly from Lemma 4. If $w\geq\frac{1}{2}$ then set $w=\frac{1}{2}$ in (4) which maximizes the lower bound. ∎

3.2 Markov Type

For a binary codeword ${\bm{x}}=x_{1}x_{2}\cdots x_{n}\in\{0,1\}^{n}$, the number of switches of ${\bm{x}}$ is equal to $|\{i:1\leq i\leq n-1,x_{i}\oplus x_{i+1}=1\}|$, that is the number of length-$2$ consecutive subsequence $01$ or $10$. (Here $\oplus$ is the XOR operation.) Now we present another lower bound for $C(d,\delta)$ based on the following observation.

Fact 1**.**

Let $S$ be the collection of all codewords in the Hamming cube $\{0,1\}^{n}$ that has at most $dn$ switches. Then the VC-dimension of $S$ or any subset of $S$ is at most $dn+1$.

Proof.

Let $I$ be any $dn+2$ coordinates. Let ${\bm{c}}$ be a length-($dn+2$) vector such that $c_{i}=0$ for odd $i\in\{1,2,\dots,dn+2\}$ and $c_{i}=1$ for even $i\in\{1,2,\dots,dn+2\}$. Then the number of switches of ${\bm{c}}$ is $dn+1$. Hence the projection of $S$ onto these coordinates $S|_{I}$ does not contain ${\bm{c}}$, therefore $S$ does not shatter $I$. This concludes our proof. ∎

We refer to an $(S,M,\delta n)$-code as a subset of $S$ with size $M$ and minimum distance at least $\delta n$. We will prove a GV-type bound for such $(S,M,\delta n)$-codes, and thus get a lower bound for $C(d,\delta)$. Our proof relies on a generalized GV bound provided by Kolesnik and Krachkovsky [11], and follows the same line of reasoning as in Sections III-V of [13], where Marcus and Roth developed an improved GV bound for constrained systems based on stationary Markov chains.

Lemma 6**.**

[11*, Lemma 1]**

Let $S$ be a subset of $\{0,1\}^{n}$. Then there exists an $(S,M,\delta n)$-code such that*

[TABLE]

where

[TABLE]

In order to compute our lower bound, we shall consider stationary Markov chains on graphs. A labeled graph $G=(V_{G},E_{G},L_{G})$ is a finite directed graph with vertices $V_{G}$, edges $E_{G}$, and a labeling $L_{G}:E_{G}\rightarrow\Sigma$ for some finite alphabet $\Sigma$. For any vertex $u$, the set of outgoing edges from $u$ is denoted by $E^{+}_{G}(u)$, and the set of incoming edges to $u$ is $E^{-}_{G}(u)$. A graph $G$ is called irreducible if there is a path in each direction between each pair of vertices of the graph. The greatest common divisor of the lengths of cycles of a graph $G$ is called the period of G. An irreducible graph $G$ with period $1$ is called primitive. A stationary Markov chain on a finite directed graph $G$ is a function $P:E_{G}\rightarrow[0,1]$ such that

1. (i)

$\sum_{e\in E_{G}}P(e)=1$; 2. (ii)

$\sum_{e\in E_{G}^{+}(u)}P(e)=\sum_{e\in E_{G}^{-}(u)}P(e)$ for every $u\in V_{G}$.

Evidently, $P(e)$ represents the probability that the chain will make a transition along the edge $e$. We denote by ${\mathcal{M}}(G)$ the set of all stationary Markov chains on $G$. For a stationary Markov chain $P\in{\mathcal{M}}(G)$, we introduce two dummy random variables $X,Y$ such that their joint distribution is defined by

[TABLE]

Then the condition (ii) amounts to saying that the marginal distributions of $X$ and $Y$ are equal.

For a stationary Markov chain $P\in{\mathcal{M}}(G)$ and a function $f:E_{G}\rightarrow\mathbb{R}^{k}$, we denote by ${\bf E}_{P}(f)$ the expected value of $f$ with respect to $P$, that is,

[TABLE]

Fix a vertex $u$, and let $\Gamma_{n}(G)$ denote the set of all cycles in $G$ of length $n$ that start and end at $u$. For a cycle $\gamma=e_{1}e_{2}\dots e_{n}\in\Gamma_{n}(G)$, let $P_{\gamma}$ denote the stationary Markov chain defined by

[TABLE]

We refer to $P_{\gamma}$ as the empirical distribution of the cycle $\gamma$, and to

[TABLE]

as the empirical average of $f$ on the cycle $\gamma$. (Note that the empirical distribution $P_{\gamma}$ is closely related to the so-called “second-order type” of sequence $L_{G}(e_{1})L_{G}(e_{2})\cdots L_{G}(e_{n})$.) For a subset $U\subset\mathbb{R}^{k}$, let ${\mathcal{M}}(G;f,U)$ denote the set of all stationary Markov chains $P$ on $G$ such that ${\bf E}_{P}(f)\in U$, and let

[TABLE]

The following lemma is a consequence of well-known results on second-order types of Markov chains, cf. Boza [2], Davisson, Longo, Sgarro [5], Natarajan [16], Csiszár, Cover, Choi [4], and Csiszár [3]. (Throughout this paper, the base of the logarithm is $|\Sigma|$.)

Lemma 7**.**

[13*, Lemma 2]**

Let $G$ be a primitive graph and $f:E_{G}\rightarrow\mathbb{R}^{k}$ be a function on the edges of $G$. Let $U$ be an open and nonempty subset of $\mathbb{R}^{k}$. Then*

[TABLE]

Hereafter we will consider the labeled graph $G$ over alphabet $\Sigma=\{0,1\}$ depicted in Figure 1. The labeling $L_{G}$ is defined by $L_{G}((a,b))=L_{G}((b,b))=0$ and $L_{G}((b,a))=L_{G}((a,a))=1$. On the other hand, the function $f:E_{G}\rightarrow\mathbb{R}$ is defined by $f((a,a))=f((b,b))=0$ and $f((a,b))=f((b,a))=1$. Then we can verify the following.

Fact 2**.**

For a cycle $\gamma=e_{1}e_{2}\cdots e_{n}\in\Gamma_{n}(G)$, the value $nE_{P_{\gamma}}(f)-f(e_{1})$ is equal to the number of switches of the corresponding binary sequence $L_{G}(e_{1})L_{G}(e_{2})\cdots L_{G}(e_{n})$.

Now we come to our second lower bound for $C(d,\delta)$. We will consider the subset

[TABLE]

By definition, for any ${\bm{x}}\in S_{n}(d)$ its number of switches is at most $dn$.

In order to use Lemma 6, we introduce the graph $G\times G$ whose vertex set is $V_{G\times G}=V_{G}\times V_{G}=\{\langle u,u^{\prime}\rangle:u,u^{\prime}\in V_{G}\}$ and edge set is $E_{G\times G}=E_{G}\times E_{G}=\{\langle e,e^{\prime}\rangle:e,e^{\prime}\in E_{G}\}$. Given the function $f$ defined on the edges of $G$, we define two functions $f^{(1)}$ and $f^{(2)}$ on $E_{G\times G}$ by

[TABLE]

and a function $\Delta:E_{G\times G}\rightarrow\mathbb{R}$ by

[TABLE]

Note that the function $\Delta$ is used to count the Hamming distance between two binary sequences. We collect $f^{(1)},f^{(2)}$ and $\Delta$ to define a function $\varphi:E_{G\times G}\rightarrow\mathbb{R}^{3}$ by $\varphi=[f^{(1)},f^{(2)},\Delta]$. For a subset $U\subset[0,1]$ we set

[TABLE]

In particular, we use ${\mathcal{F}}(p)$ and ${\mathcal{G}}(p,\delta)$ as short-hand notations for ${\mathcal{F}}(\{p\})$ and ${\mathcal{G}}(\{p\},\delta)$ respectively, where $0\leq p\leq 1$. Set

[TABLE]

Lemma 8**.**

There exist $(S_{n}(d),M,\delta n)$-codes satisfying

[TABLE]

Proof.

For $p\in[0,d]$ and $\varepsilon>0$, let $U_{p,\varepsilon}=(p-\varepsilon,p+\varepsilon)$,

[TABLE]

and

[TABLE]

By Lemma 7,

[TABLE]

and

[TABLE]

Note that both $H_{P}(Y|X)$ and ${\bf E}_{P}(f)$ are continuous in $P$. So if we let $\varepsilon\rightarrow 0$, then by Lemma 6 there exist $(S_{n}(d),M,\delta n)$-codes satisfying

[TABLE]

Then our result follows. ∎

Theorem 3**.**

$C(d,\delta)\geq R_{MA}(d,\delta).$**

Proof.

This follows from Lemma 8 and the fact that any $(S_{n}(d),M,\delta n)$ code has VC-dimension at most $dn+1$. ∎

Using convex duality we can compute $R_{MA}(d,\delta)$ through an unconstrained optimization problem with convex objective function as follows. For a function $f:E_{G}\rightarrow\mathbb{R}^{k}$, let $A_{G;f}({\bm{x}}),{\bm{x}}\in\mathbb{R}^{k}$, be the matrix function indexed by the states of $G$ with entries

[TABLE]

and let $\lambda_{G;f}({\bm{x}})$ denote the spectral radius of $A_{G;f}({\bm{x}})$. (Here the $\cdot$ operator in the exponent is the inner product of two vectors.) Recall the definitions of $f,f^{(1)},f^{(2)},\Delta,\varphi$, and define $\varphi^{\prime}=[f^{(1)}+f^{(2)},\Delta]:E_{G\times G}\rightarrow\mathbb{R}^{2}.$ Let $G$ be the graph of Figure 1. Then

[TABLE]

and

[TABLE]

Through direct computations, we have $\lambda_{G;f}(x)=2^{-x}+1,$ and

[TABLE]

From the well-known results in convex duality principle, we can obtain the following. Similar results are also obtained in [10, 12].

Lemma 9**.**

[13*, Lemma 5]**

Let $G$ be a graph and let $f:E_{G}\to\mathbb{R}^{k},g:E_{G}\to\mathbb{R}^{l}$ be functions on the edges of $G$. Set $\phi=[f,g]:E_{G}\to\mathbb{R}^{k+l}$. Then for any ${\bm{r}}\in\mathbb{R}^{k}$ and ${\bm{s}}\in\mathbb{R}^{l}$,*

[TABLE]

Theorem 4**.**

[TABLE]

Proof.

Applying Lemma 9 to compute ${\mathcal{F}}(p)$, we have

[TABLE]

Similarly, we have

[TABLE]

On the other hand, for $\varepsilon>0$, choose some point $(x^{\prime},y^{\prime},z^{\prime})$ such that

[TABLE]

and let $\bar{x}=(x^{\prime}+y^{\prime})/2$. Note that $\lambda_{G\times G;\varphi}(x,y,z)=\lambda_{G\times G;\varphi}(y,x,z)$ and the function $\log{\lambda_{G\times G;\varphi}(x,y,z)}$ is convex (see [13, Remark 2]). Thus

[TABLE]

and ${\mathcal{G}}(p,\delta)=\inf_{\begin{subarray}{c}x\in\mathbb{R}\\ z\in\mathbb{R}_{\geq 0}\end{subarray}}\{2px+\delta z+\log{\lambda_{G\times G;\varphi^{\prime}}(x,z)}\}$. This concludes our proof. ∎

4 Examples

Example 1**.**

We plot the bounds for $d=\frac{1}{4}$ and $\frac{1}{16}$ in Fig. 2. Note that all these bounds intersect at $R=h(d)$ when $\delta=0$; and our shortening upper bound (Thm. 1) is always better than the second MRRW bound (hence we do not plot it here). As we can see, for $d=\frac{1}{4}$ our shortening upper bound (Thm. 1) is always better than Haussler’s upper bound (Lem. 3), and the constant weight lower bound (Thm. 2) is always better than the Markov type lower bound (Thm. 3). For $d=\frac{1}{16}$, the performance of these bounds are quite different.

Example 2**.**

We plot the bounds for $\delta=\frac{1}{4}$ and $\frac{1}{16}$ in Fig. 3.

Remark 1**.**

Similarly as in [13], we can slightly improve the lower bounds by considering subsets $T$ of our chosen set $S$. For example, when $d=1/16$ and $\delta=0.1927$, both Theorem 2 and Theorem 3 give that $C(d,\delta)\geq 0.046$. On the other hand, let $T$ be the collection of all codewords in the Hamming cube $\{0.1\}^{n}$ that has weight $0.5n$ and at most $1/16n$ switches, then the generalized GV bound for subset $T$ shows that $C(d,\delta)\geq 0.0461$. **

5 Discussion

In this paper, we have studied the maximal size of a binary code with a given minimum distance and a given VC dimension. We gave two lower bounds, based on the idea of random GV-type constructions inside structured sets (Hamming balls, Markov types) in a way that simultaneously controls the minimum distance and the VC dimension. It may be interesting to consider other structured sets in order to improve the bounds, or to come up with a different method of construction.

Our weakest point is arguably the upper bound, which unlike the lower bounds, was derived by treating the problem of minimum distance and VC dimension separately. It stands to reason that a different argument that simultaneously controls both quantities could improve our bound. However, so far we have been unable to come up with such an argument. One reasonable line of attack could be to take the VC dimension constraint into consideration as part of an LP-type argument. However, the VC dimension constraint is global, and our attempts to embed it in the more local LP-type approach have not been fruitful. Another direction to consider is a blow-up argument: Given a code with minimum distance $\delta$, we blow-up the code to include parts of the Hamming balls of radius $\delta/2$ around each codeword. If this can be done in a controlled way such that the increase in the VC dimension can be accounted for, then the Sauer–Shelah lemma can be applied to the blown-up code. This currently appears to be difficult. Lastly, it would be interesting to see if a suitable shifting argument that somehow keeps the minimum distance in check can be used, to yield a bound in the spirit of the Sauer–Shelah lemma.

Acknowledgement

We would like to thank Ronny Roth for his helpful comments on Remark 1.