Open cells in the tables of best linear codes

What is the largest minimum distance $d$ of a linear code of length $n$ and dimension $k$ over a small field? Markus Grassl's tables give the best known lower and upper bounds. Every binary case up to length 31 is settled, but already at length 32 there are gaps that have stood for decades, and a better code is checked in a fraction of a second.

Problem

A linear $[n, k, d]$ code over the finite field $\mathbb{F}_q$ is a $k$-dimensional subspace of $\mathbb{F}_q^n$ in which every nonzero vector has at least $d$ nonzero coordinates. For each $(q, n, k)$ let $d_q(n, k)$ be the largest possible $d$. The tables at codetables.de list, for $q \le 9$ and lengths up to 256 (binary), the best known lower bound (from an explicit code) and upper bound (from a proof) on $d_q(n, k)$.

Task: close a gap in the table. The smallest open cells are:

• over $\mathbb{F}_2$: $[32, 14]$ with $8 \le d \le 9$ and $[32, 18]$ with $6 \le d \le 7$, followed by four cells of length 33;
• over $\mathbb{F}_4$: $[19, 8]$ with $8 \le d \le 9$.

Background

Brouwer maintained the classic tables of bounds on linear codes, and Grassl's site extends them with a stored construction, generated in the Magma computer algebra system, for every lower bound. Lower bounds come from explicit codes, often built from cyclic, quadratic residue or other structured codes and then modified by shortening, extending or taking subcodes. Upper bounds come from arguments such as the Griesmer bound and from computer classifications. Many entries are still open: for binary codes of length at most 80, about 40 percent of the cells have a gap.

What is known

• Binary codes: the table has no gaps for $n \le 31$; Jaffe (2000) determined the optimal binary codes of length at most 30.
• [32, 14] over $\mathbb{F}_2$: $d \ge 8$ from a subcode of the $[32, 17, 8]$ code of Cheng and Sloane (1989); $d \le 9$ from Jaffe's nonexistence results, via shortening to $[29, 11]$.
• [32, 18] over $\mathbb{F}_2$: $d \ge 6$ from a shortened and extended code; $d \le 7$, again via Jaffe's bound for $[29, 15]$.
• [19, 8] over $\mathbb{F}_4$: $d \ge 8$ from the quadratic residue code of length 19, shortened and restricted; $d \le 9$ from a one-step Griesmer argument.
• These constructions date from 2001 in the table, so the cells have been open for over twenty years.

What would count as progress

• A generator matrix for a code beating the lower bound, for example a binary $[32, 14, 9]$ or $[32, 18, 7]$ code, or a $[19, 8, 9]$ code over $\mathbb{F}_4$. Its minimum distance is found by enumerating the $q^k$ codewords, at most a few hundred thousand here, which takes well under a second.
• A proof that a cell's upper bound can be lowered, for instance that no binary $[32, 14, 9]$ code exists; exhaustive classifications should come with code that others can rerun.
• Improvements anywhere else in the tables, documented by an explicit construction.

References

1. Y. Cheng and N. J. A. Sloane, "Codes from symmetry groups, and a [32,17,8] code", SIAM Journal on Discrete Mathematics 2 (1989), 28–37. doi:10.1137/0402003
2. D. B. Jaffe, "Optimal binary linear codes of length ≤ 30", Discrete Mathematics 223 (2000), 135–155. doi:10.1016/s0012-365x(99)00135-1
3. M. Grassl, "Searching for linear codes with large minimum distance", in Discovering Mathematics with Magma, Springer (2006), 287–313. doi:10.1007/978-3-540-37634-7_13

Source

Imported from Bounds on the minimum distance of linear codes (codetables.de), binary table, curated by Markus Grassl. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.