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ARITHMETIO OPERATIONS MODULO AN IRREDUCIBLE
THE NUMBER OF IRREDUCIBLE qARY POLYNOMIALS
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binary BCH code binary codes binary polynomial binary symmetric channel block length channel errors check digits code of block code of length code of rate code’s codewords of weight column compute convolutional codes correct cyclic code cyclic shifts decoding algorithm deﬁned deﬁnition divides double-error-correcting dual encoder error locations error patterns error-correcting example extended BCH code ﬁeld element ﬁnd ﬁnite ﬁeld ﬁrst function GF(p given Hamming code Hamming weight iﬁ immediate subordinate irreducible binary irreducible factors irreducible polynomial Lee metric Lemma linear code message digits minimal polynomial minimum distance minimum weight multiple negacyclic nomial nonzero nth root obtained original code parity parity-check matrix permutation PG code polynomial of degree prime primitive projective subspace Proof q-ary QR code quadratic received word RM code root of unity rows satisﬁes sequence shown in Fig subcode threshold decoding vector weight enumerator words of weight zero