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    Z4-codes and their gray map images as orthogonal arrays and t-designs 
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    Z4-codes and their gray map images as orthogonal arrays and t-designs

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    KUWATAMechanismsofInterlaminar2010.PDF (21.97Mb)
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    Abstract
    This thesis discusses various connections between codes over rings, in par- ticular linear Z4-codes and their Gray map images as orthogonal arrays and t-designs. It also introduces the connections between VC-dimension of binary codes and the strengths of the codes as orthogonal arrays. The second chapter concerns codes over rings. It is known that if we have a matrix A over a eld F, whose rows form a linear code, such that any t columns of A are linearly independent then A is an orthogonal array of strength t. I shall begin with generalising this theorem to any nite commutative ring R with identity. The case R = Z4 is particularly important, because of the Gray map, an isometry from Zn 4 (with Lee weight) to Z2n 2 (with Hamming weight). I determine further connections that exist between the strength of a linear code C over Z4 as an orthogonal array, the strength of its Gray map image as an orthogonal array and the minimum Hamming and Lee weights of its dual C?. I also nd that the strength of a binary code as an orthogonal array is less than or equal to its strong VC-dimension. The equality holds for linear binary codes. Furthermore, the lower bound is also determined for the strength of the Gray map image of any linear Z4-code. 4 Moreover, I show that if a code over any alphabet is an orthogonal array with a certain constraint then the supports of the codewords of some Hamming weight form a t-design. Furthermore, I prove that if a linear Z2- code satis es the t-mixture condition, then such a code is an orthogonal array of strength t. I then investigate if such connection also exists for non- linear Gray map images of linear Z4-codes, and prove that it does for some values t.
    Authors
    Kusuma, Josephine
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    https://qmro.qmul.ac.uk/xmlui/handle/123456789/566
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    • Theses [3702]
    Copyright statements
    The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author
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