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arXiv · 1503.03394

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
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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.

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Michael Kiermaier, Alfred Wassermann, Johannes Zwanzger. 2015-03-10. New upper bounds on binary linear codes and a $\mathbb Z_4$-code with a better-than-linear Gray image. https://doi.org/10.1109/tit.2016.2612654

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