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Quasi-cyclic codes of index 2 and skew polynomial rings over finite fields. (English) Zbl 1292.94176

Summary: Let \(\theta\) be the Frobenius automorphism of the finite field \(\mathbb F_{q^l}\) over its subfield \(\mathbb F_q\), \(\mathbb F_{q^l}[Y, \theta]\) the skew polynomial ring and \(\mathbb F_{q^l}[Y, \theta]/\langle Y^l-1\rangle\) the quotient ring of \(\mathbb F_{q^l}[Y, \theta]\) modulo its ideal \(\langle Y^l-1\rangle\). We construct a specific \(\mathbb F_q\)-algebra isomorphism from \(\mathbb F_{q^l}[Y, \theta]/\langle Y^l-1\rangle\) onto the matrix ring \(M_l(\mathbb F_q)\), and investigate factorizations of polynomials in \(\mathbb F_q[X]\) over \(\mathbb F_q\) and \(\mathbb F_{q^l}\) when \(l\) is a prime integer. Then we present an algorithm to calculate monic factors of \(X^m-1\) in \(\mathbb F_{q^2}[Y,\theta]/\langle Y^2-1\rangle[X]\), and construct a class of quasi-cyclic codes of length \(2m\) and index 2 over \(\mathbb F_q\) from these monic factors by use of an \(\mathbb F_q\)-algebra isomorphism from \(\mathbb F_{q^2}[Y,\theta]/\langle Y^2-1\rangle[X]\) onto \(M_2(\mathbb F_q)[X]\).

MSC:

94B05 Linear codes (general theory)
94B15 Cyclic codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
16S36 Ordinary and skew polynomial rings and semigroup rings
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References:

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