A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. © 2013 IEEE.
|Journal||IEEE Transactions on Information Theory|
|Publication status||Published - 29 Jul 2013|
- 1-perfect codes
- BBZ BBZ Q -codes 2 4 8
- Hadamard codes
- propelinear codes
- translation invariant codes