On $\mathbb{Z}_{\text{8}}$ -Linear Hadamard Codes: Rank and Classification

Abstract

The Z2s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2slinear Hadamard codes of length 2 t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.

Equivalences among Z2s-linear Hadamard codes

“…Tables 1 and 3 where r is the rank (computed by using the computer algebra system Magma [4,20]) and k is the dimension of the kernel ( [7,14]). The rank for s = 2 and s = 3 can also be computed by using the results given in [18] and [8], respectively. Note that if two codes have different values (r, k), then they are not equivalent.…”

“…From [8], we have that, considering only the rank, it is not possible to fully classify these codes either.…”

“…If this equation is also true for any t ≥ 12, then upper bounds (3) and (4) coincide. Moreover, upper bound (14) would always be better than (15) sinceX t,s ≤ X t,s − 2 = A t,s − 1. given in [14] and [8], respectively. By the results in Section 3, we know that In Table 6, we can see the complete classification of the Z 2 s -linear Hadamard codes with s ∈ {2, 3} for ≤ t ≤ 9.…”

“…For s > 2, the dimension of the kernel for Z 2 s -linear Hadamard codes of length 2 t is established in [7], and it is proved that this invariant only provides a complete classification for certain values of t and s. Lower and upper bounds are also established for the number of nonequivalent Z 2 s -linear Hadamard codes of length 2 t , when both t and s are fixed, and when just t is fixed; denoted by A t,s and A t , respectively. The rank of these codes is computed in [8] only for s = 3, and it is proved that in this case the rank is not enough to obtain a complete classification. However, it is also shown that, for s = 3, by using both invariants, the rank and dimension of the kernel, it is possible to provide a full classification for any t ≥ 3.…”

“…Moreover, the exact value of A t,3 is given. From [7] and [8], we can check that there are nonlinear codes having the same rank and dimension of the kernel for different values of s, once the length 2 t is fixed, for all 5 ≤ t ≤ 11.…”