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Characterization theorems for extending Goppa codes to cyclic codes (Corresp.)

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2 Author(s)

Several theorems are presented which characterize Goppa codes having the property of becoming cyclic when an overall parity cheek is added. If such a Goppa code has location set L = GF (q^{m}) and a Goppa polynomial g(z) that is irreducible over GF(q^{m}) , then g(z) must be a quadratic. Goppa codes defined by (z- \beta )^{a} and location set L with cardinality n such that n+l|q^{m}-1 are considered along with their subcodes. A sufficient condition on L is derived for the extended codes to become cyclic. This condition is also necessary when a = 1. The construction of L for different n satisfying the stated condition is investigated in some detail. Some irreversible Goppa codes have been shown to become cyclic when extended by an overall parity check.

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IEEE Transactions on Information Theory  (Volume:25 ,  Issue: 2 )