New upper bounds on binary linear codes and a $\mathbb Z_4$-code with a better-than-linear Gray image
Michael Kiermaier, Alfred Wassermann, Johannes Zwanzger
TL;DR
The paper proves there is no binary linear $[1988,12,992]$ code, by combining exact MacWilliams equations, residual-code arguments, Griesmer bounds, table lookups, and a lattice-based ILP solver. This also yields nonexistence results for $[324,10,160]$, $[356,10,176]$, $[772,11,384]$, and $[836,11,416]$ binary linear codes. The motivation arises from the ${ m Z}_4$-linear extended dualized Kerdock code ${\hat{\mathcal{K}}^*_{6}}$, whose binary Gray image is $(1988,2^{12},992)$ and is shown to be BTl, i.e., better than any analogous binary linear code. The work demonstrates the power of combining exact algebraic constraints with ILP and table lookups to derive nonexistence results for otherwise challenging parameter sets and situates ${\hat{\mathcal{K}}^*_{6}}$ among the small set of known BTl ${\mathbb Z}_4$-code Gray images. It also leaves open the question of whether further codes in the ${\hat{\mathcal{K}}^*_{k+1}}$ series share the BTl property.
Abstract
Using integer linear programming and table-lookups we prove that there is no binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$, $[772,11,384]$, and $[836,11,416]$ is shown. Our work is motivated by the recent construction of the extended dualized Kerdock code $\hat{\mathcal{K}}^*_{6}$, which is a $\mathbb{Z}_4$-linear code having a non-linear binary Gray image with the parameters $(1988,2^{12},992)$. By our result, the code $\hat{\mathcal{K}}^*_{6}$ can be added to the small list of $\mathbb{Z}_4$-codes for which it is known that the Gray image is better than any binary linear code.