TY - JOUR
T1 - Families of hadamard Z2Z4Z8-codes
AU - Del Rio, Angel
AU - Rifa, Josep
PY - 2013/7/29
Y1 - 2013/7/29
N2 - 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.
AB - 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.
KW - 1-perfect codes
KW - BBZ BBZ Q -codes 2 4 8
KW - Hadamard codes
KW - propelinear codes
KW - translation invariant codes
U2 - 10.1109/TIT.2013.2258373
DO - 10.1109/TIT.2013.2258373
M3 - Article
SN - 0018-9448
VL - 59
SP - 5140
EP - 5151
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 8
M1 - 6508950
ER -