Independence of vectors in codes over rings

Abstract

We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors.

Do non-free LCD codes over finite commutative Frobenius rings exist?

“…All three factors of X 9 − 1 over 8 are self-reciprocal polynomials in 8 [X ] and hence all cyclic codes of length 9 over 8 are LCD and so reversible.…”

“…[9, 4 2 , 6] Good (X 6 + X 3 + 1) [9, 4 3 , 3] Good (X − 1)(X 2 + X + 1) [9, 4 6 , 2] Good (X 8 + X 7 + 3X 6 + 3X 4 + 3X 2 + X + 1) [17, 4 9 , 7] Optimal (X − 1)(X 8 [31, 4 21 , 6] Optimal g 2 g 3 g 4 g 5 g 6 g 7 g 9 g 10 g 12 g 13 [63, 4 13 , 36] Optimal g 1 g 3 g 4 g 5 g 6 g 7 g 9 g 10 g 12 g 13 [63, 4 14 , 34] Optimal g 3 g 4 g 6 g 7 g 8 g 10 g 11 g 12 g 13 [63, 4 15 , 21] Optimal g 1 g 6 g 7 g 10 g 11 g 12 g 13 [63, 4 20 , 18] Optimal g 1 g 5 g 6 g 7 g 9 g 13 [63, 4 32 , 16] Optimal g 1 g 2 g 7 g 8 g 10 g 11 g 12 g 13 [63, 4 24 , 14] Optimal g 2 g 3 g 4 g 6 g 10 g 12 [63, 4 37 , 12] Optimal g 1 g 2 g 3 g 4 g 10 g 12 [63, 4 42 , 10] Optimal g 2 g 5 g 6 g 7 g 9 g 13 [63, 4 31 , 9] Optimal g 3 g 4 g 7 g 8 g 11 g 13 [63, 4 33 , 7] Optimal g 1 g 8 g 11 [63, 4 50 , 6] Optimal Example 4.4. The factorization of X 15 − 1 over 8 into a product of basic irreducible polynomials over 8 is given by…”

Self-dual codes over commutative Frobenius rings

“…we know that (see [8]) R is a finite local ring, and its maximal ideal is m = 2, x = 2 + x = {2a + bx | a, b ∈ Z 4 }. The 16 elements in R are as follows:…”

Codes over Rk, Gray maps and their binary images

“…For codes over rings, the situation is different. (See [3] and [5] for example.) In [3], the following definitions were given.…”

“…It is proven in [3] that any linear code has a minimal set of vectors that generate the code that is both modular independent and independent. We shall call such a set a basis for the code.…”