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Bounds on $k$-hash distances and rates of linear codes

Stefano Della Fiore, Marco Dalai

TL;DR

This work studies upper and lower bounds on the rate of linear $q$-ary codes in ${\mathbb F}_q^n$ with minimum $k$-hash distance $d_k$, introducing a general framework that relates $d_k$ to the usual Hamming distance and to the code rate. It first treats the special case $q=k=3$, providing an achievability proof via a tetracode-based construction and extending to $d_3>1$, and derives improved linear-code upper bounds by extending Jamison–Bruen type hyperplane-covering arguments; notably, it recovers the best known $d_3=1$ bound with a simpler proof and yields new bounds for $d_3>1$. The authors then present a general method for all $q\ge k\ge 3$ that yields a recursive bound on $d_k$ in terms of $d_2$, which translates into a rate bound $R \le \frac{1}{\sum_{i=1}^{k-2} \frac{(q-1)^i}{(q-2)^{\underline{i}}}} \left(\delta_2 - \frac{(q-1)^{k-2}}{(q-2)^{\underline{k-2}}}\delta_k\right) + o(1)$, with refinements when combining with LP bounds. The paper also discusses the implications for zero-error capacity under list decoding and presents a case study (the typewriter channel) where the method does not improve the known upper bound, illustrating both the reach and the limits of the approach. Overall, the results advance the understanding of the rate–distance trade-offs for linear hash-like codes and connect to broader zero-error and list-decoding problems.

Abstract

In this paper, we bound the rate of linear codes in $\mathbb{F}_q^n$ with the property that any $k\leq q$ codewords are all simultaneously distinct in at least $d_k$ coordinates. For the case of particular interest $q=k=3$ we recover, with a simpler proof, state of the art results in the case $d_3=1$ and new bounds for $d_3>1$. We finally discuss some related open problems on the list-decoding zero-error capacity of discrete memoryless channels.

Bounds on $k$-hash distances and rates of linear codes

TL;DR

This work studies upper and lower bounds on the rate of linear -ary codes in with minimum -hash distance , introducing a general framework that relates to the usual Hamming distance and to the code rate. It first treats the special case , providing an achievability proof via a tetracode-based construction and extending to , and derives improved linear-code upper bounds by extending Jamison–Bruen type hyperplane-covering arguments; notably, it recovers the best known bound with a simpler proof and yields new bounds for . The authors then present a general method for all that yields a recursive bound on in terms of , which translates into a rate bound , with refinements when combining with LP bounds. The paper also discusses the implications for zero-error capacity under list decoding and presents a case study (the typewriter channel) where the method does not improve the known upper bound, illustrating both the reach and the limits of the approach. Overall, the results advance the understanding of the rate–distance trade-offs for linear hash-like codes and connect to broader zero-error and list-decoding problems.

Abstract

In this paper, we bound the rate of linear codes in with the property that any codewords are all simultaneously distinct in at least coordinates. For the case of particular interest we recover, with a simpler proof, state of the art results in the case and new bounds for . We finally discuss some related open problems on the list-decoding zero-error capacity of discrete memoryless channels.
Paper Structure (6 sections, 11 theorems, 57 equations, 6 figures, 1 table)

This paper contains 6 sections, 11 theorems, 57 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The case $q=k=3$
  3. Lower Bound
  4. Upper Bound
  5. General Case $q\geq k\geq 3$
  6. A Related Open Problem in Zero-Error Capacity

Key Result

Theorem 1

For any $\delta_3>0$, ternary codes with asymptotic relative $3$-hash distance $\delta_3$ exist for all rates $R$ satisfying where $D(\cdot \| \cdot)$ is the Kullback-Leibler divergence cover-thomas-book (with $\log$ in base $3$), $p$ is the probability vector $p=(\frac{25}{81},\frac{48}{81},0,\frac{8}{81}, 0)$ and $p^*$ is defined by with $\alpha$ such that $\sum_{j}j\cdot p_j^*=4\delta_3$.

Figures (6)

  • Figure 1: Achievable rates for linear ternary codes with $3$-hash distance $\delta_3$.
  • Figure 2: Bounds on the rate of linear $7$-ary codes with $4$-hash distance $\delta_4$.
  • Figure 3: Bounds on the rate of $q$-ary linear $4$-hash codes for $q \geq 5$.
  • Figure 4: Channel with a positive zero-error list-decoding capacity with list-size $2$; no output is compatible with more than two inputs. Zero-error codes for these two channels are, respectively, $(3,3)$-hash (trifferent) and $(4,3)$-hash codes.
  • Figure 5: On the left, a $5$-input typewriter channel. On the right, its confusability graph. Zero-error codes under list-decoding with list-size $2$ correspond in this case to triangle-free subgraphs of the strong power of the pentagon.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1: jamison-1977
  • Lemma 2: bruen-1992
  • Theorem 2
  • Lemma 3
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • ...and 2 more