A class of binary cyclic codes with five weights

Abstract

In this paper, the dual code of the binary cyclic code of length 2 n − 1 with three zeros α, % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-J % bba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaae % qabaWaaqaafaaakeaarmqr1ngBPrgitLxBI9gBaGGbciab-f7aHnaa % CaaaleqabaacbiGae4hDaq3aaSbaaWqaaGqaaiab9fdaXaqabaaaaa % aa!45E9! $$ \alpha ^{t_1 } $$ and % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-J % bba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaae % qabaWaaqaafaaakeaarmqr1ngBPrgitLxBI9gBaGGbciab-f7aHnaa % CaaaleqabaacbiGae4hDaq3aaSbaaWqaaGqaaiab9jdaYaqabaaaaa % aa!45EB! $$ \alpha ^{t_2 } $$ is proven to have five nonzero Hamming weights in the case that n ⩾ 4 is even and t 1 = 2n/2 + 1, t 2 = 2n−1 − 2n/2−1 + 1 or 2n/2 + 3, where α is a primitive element of the finite field % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-J % bba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaae % qabaWaaqaafaaakeaatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0Hgi % uD3BaGGbaiab-vi8gnaaBaaaleaaieaacqGFYaGmdaahaaadbeqaaG % qaciab95gaUbaaaSqabaaaaa!4B25! $$ \mathbb{F}_{2^n } $$ . The dual code is a divisible code of level n/2 −1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward’s bound.