Automated Discovery of Improved Constant Weight Binary Codes

TL;DR

Improved lower bounds on A(n,d,w) are established by constructing new larger codes, for 24 values of $(n,d,w) with $6 \leq d \leq 18$ and $18 \leq n \leq 35$.

Abstract

A constant weight binary code consists of $n$-bit binary codewords, each with exactly $w$ bits equal to 1, such that any two codewords are at least Hamming distance $d$ apart. $A(n,d,w)$ is the maximum size of a constant weight binary code with parameters $n,d,w$. We establish improved lower bounds on $A(n,d,w)$ by constructing new larger codes, for 24 values of $(n,d,w)$ with $6 \leq d \leq 18$ and $18 \leq n \leq 35$. The improved lower bounds come from two strategies. The first is a tabu search that operates at the level of bit swaps. The second is a novel greedy heuristic that repeatedly chooses the candidate codeword that maximizes a randomly-scored histogram of distances to previously-added codewords. These strategies were proposed by CPro1, an automated protocol that generates, implements, and tests diverse strategies for combinatorial constructions.

Figures (8)

• Figure 1: The problem definition used with CPro1.
• Figure 2: CPro1's prompt for listing 20 candidate strategies. This is appended to the problem definition in Fig. \ref{['fig:problemdef']}. Half the runs here included the bold sentence of the prompt requesting strategies with a novel element; the other half use the default configuration of CPro1 which omits this extra sentence.
• Figure 3: Bit-Swap Tabu Search. In the efficient implementation, Conflict-Pairs and Penalty are maintained incrementally. Note HD($x,y$) is the Hamming Distance between binary words $x$ and $y$. We set $t_{min}=5$, $t_{max}=15$, max_no_improve_restart=1M.
• Figure 4: Random-Score Distance Histograms (RSDH). $n,w,d$ are the parameters of the code, and $s$ is the target size (number of codewords). Note HD($x,y$) is the Hamming Distance between binary words $x$ and $y$.
• Figure 5: Sliced Random-Score Distance Histograms (S-RSDH). $b$ is the number of bits that define the slice; we use $b \in \{1,2,3\}$. $t$ is the number of trials for the remainder of the code; we fix $t=1000$. Greedy-Build, Rand-Vec, and HD are defined in Fig. \ref{['fig:rsdhpseudocode']}.