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Sharper upper bounds for $q$-ary and constant-weight $B_2$ codes

Stefano Della Fiore

Abstract

We derive refined entropy upper bounds for $q$-ary $B_2$ codes by exploiting the Fourier structure of the i.i.d. difference distribution $D=X-Y$. Since the pmf of $D$ is an autocorrelation, its Fourier series is a nonnegative trigonometric polynomial of degree at most $q-1$. This leads to a natural convex relaxation over candidate difference distributions, equivalently expressible through an infinite family of positive semidefinite Toeplitz constraints. The resulting formulation admits a simple Gram interpretation and yields certified upper bounds through truncated semidefinite programs. Combined with the prefix-suffix method, this gives improved asymptotic rate upper bounds for $q$-ary $B_2$ codes; in particular, for $q\in\{9,10,11,12,13\}$ the resulting values improve on the best bounds known in the literature. We also study binary constant-weight $B_2$ codes. Extending the distance-distribution method of Cohen, Litsyn, and Zémor to the constant-weight setting, and combining it with Litsyn's asymptotic linear-programming bound for constant-weight codes, we derive a new upper bound on the constant-weight $B_2$ rate.

Sharper upper bounds for $q$-ary and constant-weight $B_2$ codes

Abstract

We derive refined entropy upper bounds for -ary codes by exploiting the Fourier structure of the i.i.d. difference distribution . Since the pmf of is an autocorrelation, its Fourier series is a nonnegative trigonometric polynomial of degree at most . This leads to a natural convex relaxation over candidate difference distributions, equivalently expressible through an infinite family of positive semidefinite Toeplitz constraints. The resulting formulation admits a simple Gram interpretation and yields certified upper bounds through truncated semidefinite programs. Combined with the prefix-suffix method, this gives improved asymptotic rate upper bounds for -ary codes; in particular, for the resulting values improve on the best bounds known in the literature. We also study binary constant-weight codes. Extending the distance-distribution method of Cohen, Litsyn, and Zémor to the constant-weight setting, and combining it with Litsyn's asymptotic linear-programming bound for constant-weight codes, we derive a new upper bound on the constant-weight rate.

Paper Structure

This paper contains 9 sections, 6 theorems, 73 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Fourier representation of the difference distribution
  3. Entropy bounds via Fourier positivity and Toeplitz Gram matrices
  4. Fourier-analytic relaxation
  5. Toeplitz formulation and Gram interpretation
  6. Improved bounds for $B_2$ codes
  7. Numerical evaluation and comparison with known bounds
  8. A Cohen--Litsyn upper bound for constant-weight $B_2$ codes
  9. Comparison with the best previously known bound

Key Result

Lemma 3.2

For every $q\ge 2$,

Figures (1)

  • Figure 1: Comparison of upper bounds on the asymptotic rate of binary constant-weight $B_2$ codes. The curve labeled "Ours" corresponds to Theorem \ref{['thm:cw-B2-LP']}, while "Sima et al." denotes the bound $R_S(\alpha)$ from SimaLiShomoronyMilenkovic. The upper bound for general codes from CohenLitsynZemor and the constructive lower bound from SimaLiShomoronyMilenkovic are also shown for reference.

Theorems & Definitions (9)

  • Definition 1.1
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3: Toeplitz SDP formulation
  • Proposition 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 6.1