Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes
نویسندگان
چکیده
منابع مشابه
Cyclic Codes and Self-Dual Codes Over
We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u = 0 and study them by analogy with the Z4 case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z4-codes. Unlike the Z4 case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-d...
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In this paper we investigate binary formally self-dual codes as images of codes over rings using various Gray maps.
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Self-dual codes over Z2 ×Z4 are subgroups of Z2 ×Zβ4 that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values α, β such that there exist a code C ⊆ Z 2 ×Z 4 are established. Moreover, the construction of a Z2Z4-linear code for each type and possible pair (α, β) is given. Fi...
متن کاملCyclic Codes and Self-Dual Codes Over F2 + uF2
We introduce linear cyclic codes over the ring F 2 + uF 2 = f0; 1; u; u = u + 1g, where u 2 = 0. This ring shares many properties of Z 4 and F 4 and admits a linear "Gray map". Cyclic codes are described as modules over (F 2 + uF 2) n which may not be free. Self-dual codes of odd length exists as in the case of Z 4-codes. We exhibit some extremal codes of this very interesting family.
متن کاملGray Images of Constacyclic Codes over some Polynomial Residue Rings
Let be the quotient ring where is the finite field of size and is a positive integer. A Gray map of length over is a special map from to ( . The Gray map is said to be a ( )-Gray map if the image of any -constacyclic code over is a -constacyclic code over the field . In this paper we investigate the existence of ( )-Gray maps over . In this direction, we find an equivalent ...
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ژورنال
عنوان ژورنال: Advances in Mathematics of Communications
سال: 2011
ISSN: 1930-5346
DOI: 10.3934/amc.2011.5.571