35 research outputs found
On the Kernel of -Linear Hadamard Codes
The -additive codes are subgroups of ,
and can be seen as a generalization of linear codes over and
. A -linear Hadamard code is a binary Hadamard
code which is the Gray map image of a -additive code. It is
known that the dimension of the kernel can be used to give a complete
classification of the -linear Hadamard codes. In this paper, the
kernel of -linear Hadamard codes and its dimension are
established for . Moreover, we prove that this invariant only provides a
complete classification for some values of and . The exact amount of
nonequivalent such codes are given up to for any , by using
also the rank and, in some cases, further computations
Linearity and Classification of -Linear Hadamard Codes
The -additive codes are subgroups of
. A -linear
Hadamard code is a Hadamard code which is the Gray map image of a
-additive code. A recursive construction
of -additive Hadamard codes of type
with , , , , , and is
known. In this paper, we generalize some known results for
-linear Hadamard codes to
-linear Hadamard codes with , , and . First, we show for which
types the corresponding -linear Hadamard
codes of length are nonlinear. For these codes, we compute the kernel and
its dimension, which allows us to give a partial classification of these codes.
Moreover, for , we give a complete classification by
providing the exact amount of nonequivalent such codes. We also prove the
existence of several families of infinite such nonlinear
-linear Hadamard codes, which are not
equivalent to any other constructed
-linear Hadamard code, nor to any
-linear Hadamard code, nor to any previously
constructed -linear Hadamard code with , with the
same length .Comment: arXiv admin note: text overlap with arXiv:2301.0940
On Z8-linear Hadamard codes : rank and classification
The Z2s -additive codes are subgroups of â€Zn2s, and can be seen as a generalization of linear codes over â€2 and â€4. A Zs-linear Hadamard code is a binary Hadamard code which is the Gray map image of a â€s -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 â€4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of â€2s-linear Hadamard codes of length 2t 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 â„ 3 is fixed. In this case, the number of nonequivalent such codes is also established
On recursive constructions of Z2Z4Z8-linear Hadamard codes
The Z2Z4Z8-additive codes are subgroups of Z α1 2 Ă Z α2 4 Ă Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 Ìž= 0, α2 Ìž= 0, and α3 Ìž= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 â„ 1, t2 â„ 0, and t3 â„ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 Ìž= 0 and α2 Ìž= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s â„ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones
-Additive Hadamard Codes
The -additive codes are subgroups of
, and can be seen as linear codes over
when , -additive or -additive
codes when or , respectively, or
-additive codes when . A
-linear Hadamard code is a Hadamard code
which is the Gray map image of a
-additive code. In this paper, we
generalize some known results for -linear Hadamard
codes to -linear Hadamard codes with
, , and . First, we give a
recursive construction of -additive
Hadamard codes of type with
, , and . Then, we show that in general the
-linear, -linear and
-linear Hadamard codes are not included in the family
of -linear Hadamard codes with , , and . Actually, we point out that
none of these nonlinear -linear Hadamard
codes of length is equivalent to a
-linear Hadamard code of any other type,
a -linear Hadamard code, or a
-linear Hadamard code, with , of the same length
On the linearity and classification of Z_p^s-linear generalized Hadamard codes
Acord transformatiu CRUE-CSICZp^s-additive codes of length n are subgroups of (Zp^s)^n , and can be seen as a generalization of linear codes over Z2, Z4 , or Z2^s in general. A Zp^s-linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zp^s-additive code by a generalized Gray map. In this paper, we generalize some known results for Zp^s-linear GH codes with p = 2 to any odd prime p. First, we show some results related to the generalized Carlet's Gray map. Then, by using an iterative construction of Zp^s -additive GH codes of type (n; t 1 , . . . , t s ), we show for which types the corresponding Zp^s-linear GH codes of length p^t are nonlinear over Zp .For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for p â„ 3 are different from the case with p = 2. Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any p â„ 2; by using also the rank as an invariant in some specific cases
On the equivalence of Z ps -linear generalized Hadamard codes
Altres ajuts: acords transformatius de la UABLinear codes of length n over Zps , p prime, called Zps -additive codes, can be seen as subgroups of Zn ps . A Zps -linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zps -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p â„ 3 prime, we establish that some Zps -linear GH codes of length pt having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t = 10 if p = 3 or t = 8 if p = 5, this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel
Linearity and classification of ZpZp^2-linear generalized Hadamard codes
The ZpZp2 -additive codes are subgroups of Zα1 p à Zα2 p2 , and can be seen as linear codes over Zp when α2 = 0, Zp2 -additive codes when α1 = 0, or Z2Z4-additive codes when p = 2. A ZpZp2 -linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2 -additive code. Recursive constructions of ZpZp2 -additive GH codes of type (α1, α2;t1,t2) with t1,t2 ℠1 are known. In this paper, we generalize some known results for ZpZp2 -linear GH codes with p = 2 to any p ℠3 prime when α1 = 0, and then we compare them with the ones obtained when α1 = 0. First, we show for which types the corresponding ZpZp2 -linear GH codes are nonlinear over Zp. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike Z4-linear Hadamard codes, the Zp2 -linear GH codes are not included in the family of ZpZp2 - linear GH codes with α1 = 0 when p ℠3 prime. Indeed, there are some families with infinite nonlinear ZpZp2 -linearGH codes, where the codes are not equivalent to any Zps - linear GH code with s ℠2
Construction and classification of Zâs-linear Hadamard codes
The Zâs-additive and ZâZâ-additive codes are subgroups of Zâs^n and Zâ^α Ă Zâ^ÎČ, respectively. Both families can be seen as generalizations of linear codes over Zâ and Zâ. A Zâs-linear (resp. ZâZâ-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Zâs-additive (resp. ZâZâ-additive) code. It is known that there are exactly â(tâ1)/2â and ât/2â nonequivalent ZâZâ-linear Hadamard codes of length 2á”, with α=0 and αâ 0, respectively, for all tâ„3. In this paper, new Zâs-linear Hadamard codes are constructed for s>2, which are not equivalent to any ZâZâ-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Zâ-linear Hadamard codes of length 2á” are pairwise nonequivalent