The Weight Enumerator of Three Families of Cyclic Codes
Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. Howe...
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| Veröffentlicht in: | IEEE transactions on information theory 2013-09, Vol.59 (9), p.6002-6009 |
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| creator | Zhou, Zhengchun Zhang, Aixian Ding, Cunsheng Xiong, Maosheng |
| description | Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p 2m -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined. |
| doi_str_mv | 10.1109/TIT.2013.2262095 |
| format | Article |
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Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p 2m -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. 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Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p 2m -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined.</description><subject>Applied sciences</subject><subject>Codes</subject><subject>Coding theory</subject><subject>Coding, codes</subject><subject>Communications systems</subject><subject>Cyclic codes</subject><subject>Educational institutions</subject><subject>Eigenvalues and eigenfunctions</subject><subject>Exact sciences and technology</subject><subject>exponential sum</subject><subject>Hermitian forms graphs</subject><subject>Information storage</subject><subject>Information theory</subject><subject>Information, signal and communications theory</subject><subject>Linear codes</subject><subject>Polynomials</subject><subject>quadratic form</subject><subject>Signal and communications theory</subject><subject>Telecommunications and information theory</subject><subject>Vectors</subject><subject>Weight</subject><subject>weight distribution</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNo9kEFLAzEQhYMoWKt3wcuCeNyaZDNJ9iiL1ULBy4LHEJNZm7Lt1mR76L83paWnYWbeezN8hDwyOmOM1q_top1xyqoZ55LTGq7IhAGospYgrsmEUqbLWgh9S-5SWudWAOMTotoVFt8Yfldj8b7dbzDacYjF0BXtKiIWc7sJfcB0nDQH1wdXNIPHdE9uOtsnfDjXKWnn723zWS6_PhbN27J0lazGEqTntf7pFHcoNFq0VDLw3FkvONRaOiVVhRolBQZKaaDIvfIgdEeFr6bk-RS7i8PfHtNo1sM-bvNFwwSXGjhwkVX0pHJxSCliZ3YxbGw8GEbNkY7JdMyRjjnTyZaXc7BNzvZdtFsX0sXH81uqYjTrnk66gIiXtYSMT7DqH_oYaq4</recordid><startdate>20130901</startdate><enddate>20130901</enddate><creator>Zhou, Zhengchun</creator><creator>Zhang, Aixian</creator><creator>Ding, Cunsheng</creator><creator>Xiong, Maosheng</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p 2m -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. 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| subjects | Applied sciences Codes Coding theory Coding, codes Communications systems Cyclic codes Educational institutions Eigenvalues and eigenfunctions Exact sciences and technology exponential sum Hermitian forms graphs Information storage Information theory Information, signal and communications theory Linear codes Polynomials quadratic form Signal and communications theory Telecommunications and information theory Vectors Weight weight distribution |
| title | The Weight Enumerator of Three Families of Cyclic Codes |
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