通信BCH编码英文文献翻译 通信BCH编码英文文献翻译(2)Divide into .Bythedivisionalgorithmweget mod bypowersof isinthecodespaceof .Sincedegdeg, .Noticethenthat isnotafieldsinceithaszerodivisors.(3)Let correspondtoanelementin .Divide into (2)weknowthat correspondstoacodewordandbyassumptionsodoes .So,since isclosedundersubtraction, mod andhasdegreelessthan .Itfollowsthat and ,deg sodeg ,wehaveshownthattheproductofanytwopolynomials isin .Thus,thesepolynomialsyieldallofthecodewordsof .(4)By(2)wecanwrite .Let correspondtoanelementin .Thenby(3) mod Nextsupposethat .Wecannowdividethroughby andgetandso corespondstoacodewordof . Remark:Therewillbeauniquepolynomial suchthat .Whenthinkingofthecodespaceasaproductofpolynomialswith insteadoftherowspaceofabinarymatrix,thispolynomial beacodewordthen Thenif deg thecoeffiecientsofthe highestpowersmatchthecoeffiecientsofthe lowestpowersofxinthepolynomial .Further,ifthegeneratingpolynomial hasdegree ,thenthecodewords formabasisforthecodespae withageneratingmatrix andwecanrealizeacodewordasaninformationsequence timesthemat
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