New Upper Bounds on Binary Linear Codes and a $ {\mathbb Z}_{4}$ -Code With a Better-Than-Linear Gray Image

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 K
₆
*, which is a Z
₄
-linear code having a non-linear binary Gray image with the parameters 1988, 2
¹²
,992. By our result, the code K
₆
* can be added to the small list of Z
₄
-codes for which it is known that the Gray image is better than any binary linear code.