7 research outputs found
On the constructions of ZpZp2-linear generalized Hadamard codes
Altres ajuts: acord transformatiu CRUE-CSICThe ZZ-additive codes are subgroups of Z ×Z , and can be seen as linear codes over Z when α=0, Z-additive codes when α=0, or ZZ-additive codes when p=2. A ZZ-linear generalized Hadamard (GH) code is a GH code over Z which is the Gray map image of a ZZ-additive code. In this paper, we generalize some known results for ZZ-linear GH codes with p=2 to any p≥3 prime when α≠0. First, we give a recursive construction of ZZ-additive GH codes of type (α,α;t,t) with t,t≥1. We also present many different recursive constructions of ZZ-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z-linear GH codes, when p≥3 prime, the Z-linear GH codes are not included in the family of ZZ-linear GH codes with α≠0. Indeed, we observe that the constructed codes are not equivalent to the Z-linear GH codes for any s≥2
Butson full propelinear codes
In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the kth roots of unity, we can construct a larger Butson matrix over the ℓth roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ. We prove that a Zps-additive code with p a prime number is isomorphic as a group to a BH-code over Zps and the image of this BH-code under the Gray map is a BH-code over Zp (binary Hadamard code for p=2). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided
Butson full propelinear codes
In this paper we study Butson Hadamard matrices, and codes over finite rings
coming from these matrices in logarithmic form, called BH-codes. We introduce a
new morphism of Butson Hadamard matrices through a generalized Gray map on the
matrices in logarithmic form, which is comparable to the morphism given in a
recent note of \'{O} Cath\'{a}in and Swartz. That is, we show how, if given a
Butson Hadamard matrix over the roots of unity, we can construct a
larger Butson matrix over the roots of unity for any
dividing , provided that any prime dividing also divides .
We prove that a -additive code with a prime number is
isomorphic as a group to a BH-code over and the image of
this BH-code under the Gray map is a BH-code over (binary
Hadamard code for ). Further, we investigate the inherent propelinear
structure of these codes (and their images) when the Butson matrix is cocyclic.
Some structural properties of these codes are studied and examples are
provided.Comment: 24 pages. Submitted to IEEE Transactions on Information Theor
-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
An enumeration of 1-perfect ternary codes
We study codes with parameters of the ternary Hamming
code, i.e., ternary -perfect codes. The rank of
the code is defined to be the dimension of its affine span. We characterize
ternary -perfect codes of rank , count their number, and prove that
all such codes can be obtained from each other by a sequence of two-coordinate
switchings. We enumerate ternary -perfect codes of length obtained by
concatenation from codes of lengths and ; we find that there are
equivalence classes of such codes.
Keywords: perfect codes, ternary codes, concatenation, switching
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 ≥ 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 ≥ 2; by using also the rank as an invariant in some specific cases