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- W2098158374 abstract "For any nonzero element <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>c</tex> of a general finite field <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>GF(q)</tex> , it is shown that the polynomials <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>(x - c)^i, i = 0,1,2,cdots</tex> , have the weight-retaining property that any linear combination of these polynomials with coefficients in <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>GF(q)</tex> has Hamming weight at least as great as that of the minimum degree polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any prime <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</tex> greater than 2, a new extensive class of <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</tex> -ary Reed-Muller codes, 2) a new class of repeated-root cyclic codes that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of the <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>p</tex> -ary Reed-Muller codes, 4) two new classes of binary convolutional codes with large free distance derived from known binary cyclic codes, 5) two new classes of long constraint length binary convolutional codes derived from <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2^r</tex> -ary Reed-Solomon codes, and 6) a new class of <tex xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</tex> -ary repeated-root constacyclic codes with an algebraic decoding algorithm." @default.
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- W2098158374 date "1973-01-01" @default.
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- W2098158374 title "Polynomial weights and code constructions" @default.
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- W2098158374 doi "https://doi.org/10.1109/tit.1973.1054936" @default.
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