Abstract
We study the extended quadratic residue code of length 24 over <TEX>$\mathbb{Z}_3$</TEX> and its lifts to rings <TEX>$\mathbb{Z}_{3^e}$</TEX> for all e including 3-adic integers ring. We completely determine the weight enumerators of all these lifts.
Highlights
A linear code of length n over R is a R-submodule of Rn
We define an inner product on Rn by (x, y) =
For v ∈ Rn, the weight wt(v) of v is defined to be the number of nonzero components of v
Summary
The dual code C⊥ of a code C of length n is defined to be C⊥ = {y ∈ Rn | (y, x) = 0 for all x ∈ C}. Key words and phrases: quadratic residue code, code over rings, self-dual code, p-adic code, weight enumertaor. Note that 3 is a quadratic residue modulo 23. Cyclic codes Q, Q1, N , N1 of length 23 with generator polynomials. Q(x), (x − 1)Q(x), N (x), (x − 1)N (x), respectively, are called quadratic residue codes defined over Z3. We have the following well-known results on quadratic residue codes defined over the field Z3. In section we are going to lift these quadratic residue codes over Z3e and to the 3-adic integers Z3∞
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