November  2012, 6(4): 401-418. doi: 10.3934/amc.2012.6.401

$\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity

1. 

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Received  June 2011 Revised  October 2012 Published  November 2012

Hirzebruch and van der Geer attached theta functions to self-orthogonal, $C\subseteq C^{\bot}$, linear codes $C\subseteq\mathbb F_p^n$, for $p$ an odd prime, and related them to the Lee weight enumerator for the code [5, Ch. 5]. Choie and Jeong extended this result to Jacobi theta functions and provided an analytic proof of the Lee weight MacWilliams Identity for such $C$ [3]. We provide an analytic proof of the Hamming weight MacWilliams Identity for linear codes $C\subseteq\mathbb F_p^n$, generalizing the seminal result for binary codes $C\subseteq\mathbb F_2^n$ [2].
Citation: David Keyes. $\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity. Advances in Mathematics of Communications, 2012, 6 (4) : 401-418. doi: 10.3934/amc.2012.6.401
References:
[1]

K. Betsumiya and Y. Choie, Jacobi forms over totally real fields and type II codes over galois rings $GR(2^m,f)$, European J. Combin., 25 (2005), 475-486. doi: 10.1016/j.ejc.2003.01.001.

[2]

M. Broué and M. Enguehard, Polynômes des poids de certains codes et fonctions theta de certains réseaux, Ann. Scie Ecole Norm. Sup., 5 (1972), 157-181.

[3]

Y. Choie and E. Jeong, Jacobi forms over totally real fields and codes over $\mathbbF_p$, Illinois J. Math., 46 (2002), 627-643.

[4]

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups," Springer-Verlag, New York, 1999.

[5]

W. Ebeling, "Lattices and Codes: A Course Partially Based on Lectures by F. Hirzebruch," Vieweg, Braunschqeig, 1994.

[6]

S. Lang, "Algebraic Number Theory," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0296-4.

[7]

J. Leech and N. J. A. Sloane, Sphere packings and error-corrective codes, Canadian J. Math., 23 (1971), 718-745. doi: 10.4153/CJM-1971-081-3.

[8]

D. Marcus, "Number Fields," Springer-Verlag, New York, 1997.

[9]

N. J. A. Sloane, Codes over $GF(4)$ and complex lattices, J. Algebra, 52 (1978), 168-181. doi: 10.1016/0021-8693(78)90266-1.

[10]

H. M. Stark, Modular forms and related objects, in "CMS Conference Proceedings,'' 7 (1987), 421-455.

[11]

L. C. Washington, "Introduction to Cyclotomic Fields," Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

show all references

References:
[1]

K. Betsumiya and Y. Choie, Jacobi forms over totally real fields and type II codes over galois rings $GR(2^m,f)$, European J. Combin., 25 (2005), 475-486. doi: 10.1016/j.ejc.2003.01.001.

[2]

M. Broué and M. Enguehard, Polynômes des poids de certains codes et fonctions theta de certains réseaux, Ann. Scie Ecole Norm. Sup., 5 (1972), 157-181.

[3]

Y. Choie and E. Jeong, Jacobi forms over totally real fields and codes over $\mathbbF_p$, Illinois J. Math., 46 (2002), 627-643.

[4]

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups," Springer-Verlag, New York, 1999.

[5]

W. Ebeling, "Lattices and Codes: A Course Partially Based on Lectures by F. Hirzebruch," Vieweg, Braunschqeig, 1994.

[6]

S. Lang, "Algebraic Number Theory," Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0296-4.

[7]

J. Leech and N. J. A. Sloane, Sphere packings and error-corrective codes, Canadian J. Math., 23 (1971), 718-745. doi: 10.4153/CJM-1971-081-3.

[8]

D. Marcus, "Number Fields," Springer-Verlag, New York, 1997.

[9]

N. J. A. Sloane, Codes over $GF(4)$ and complex lattices, J. Algebra, 52 (1978), 168-181. doi: 10.1016/0021-8693(78)90266-1.

[10]

H. M. Stark, Modular forms and related objects, in "CMS Conference Proceedings,'' 7 (1987), 421-455.

[11]

L. C. Washington, "Introduction to Cyclotomic Fields," Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

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