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Symmetries of weight enumerators and applications to Reed-Muller codes

The first author was partially supported by PEPS - Jeunes Chercheur-e-s - 2017

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  • Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 13A50, 94B60.


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  • Table 1.  $\bar S(w_{\mathcal{RM}_2(r,m)}(x,y))$

    $r\backslash m$ 1 2 3 4 5 6 7
    0 $\infty$ $D_4$ $D_8$ $D_{16}$ $D_{32}$ $D_{64}$ $D_{128}$
    1 $\infty$ $D_4$ $S_4$ $D_{8}$ $D_{16}$ $D_{32}$ $D_{64}$
    2 $\infty$ $\infty$ $D_8$ $D_{8}$ $S_4$ $D_{4}$ $D_{8}$
    3 $\infty$ $\infty$ $\infty$ $D_{16}$ $D_{16}$ $D_{4}$ $S_4$
    4 $\infty$ $\infty$ $\infty$ $\infty$ $D_{32}$ $D_{32}$ $D_{8}$
    5 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{64}$ $D_{64}$
    6 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{128}$
     | Show Table
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    Table 2.  $\bar S(w_{\mathcal{RM}_3(r,m)}(x,y))$

    $r\backslash m$ 1 2 3 4
    0 $D_3$ $D_9$ $D_{27}$ $D_{81}$
    1 $D_3$ $C_3$ $C_9$ $C_{27}$
    2 $\infty$ $C_3$ $C_3$ $C_{3}$
    3 $\infty$ $D_9$ $C_3$
    4 $\infty$ $\infty$ $C_9$
    5 $\infty$ $\infty$ $D_{27}$ $C_{3}$
    6 $\infty$ $\infty$ $\infty$ $C_{27}$
    7 $\infty$ $\infty$ $\infty$ $D_{81}$
     | Show Table
    DownLoad: CSV

    Table 3.  $\bar S(w_{\mathcal{RM}_4(r, m)}(x, y))$

    $r\backslash m$ 1 2 3
    0 $D_8$ $D_{16}$ $D_{64}$
    1 $V_4$ $C_4$ $C_{16}$
    2 $D_8$ $\{{\rm Id}\}$ $C_4$
    3 $\infty$ $\{{\rm Id}\}$ $\{{\rm Id}\}$
    4 $\infty$ $C_4$
    5 $\infty$ $D_{16}$ $\{{\rm Id}\}$
    6 $\infty$ $\infty$ $C_4$
    7 $\infty$ $\infty$ $C_{16}$
    8 $\infty$ $\infty$ $D_{64}$
     | Show Table
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