On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

[1]  Sheridan K. Houghten,et al.  The extended quadratic residue code is the only (48, 24, 12) self-dual doubly-even code , 2003, IEEE Trans. Inf. Theory.

[2]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[3]  N. J. A. Sloane,et al.  A new upper bound on the minimal distance of self-dual codes , 1990, IEEE Trans. Inf. Theory.

[4]  P. Sole,et al.  Universal codes and unimodular lattices , 1997, Proceedings of IEEE International Symposium on Information Theory.

[5]  Michael J. Razar,et al.  Modular forms for ₀() and Dirichlet series , 1977 .

[6]  Neil J. A. Sloane,et al.  Self-Dual Codes over ${\text{GF}}( 3 )$ , 1976 .

[7]  Richard A. Brualdi,et al.  Weight enumerators of self-dual codes , 1991, IEEE Trans. Inf. Theory.

[8]  John H. Conway,et al.  On the Enumeration of Self-Dual Codes , 1980, J. Comb. Theory, Ser. A.

[9]  Mark Gaulter,et al.  Minima of Odd Unimodular Lattices in Dimension 24m , 2001 .

[10]  Masaaki Harada,et al.  Type II Codes, Even Unimodular Lattices, and Invariant Rings , 1999, IEEE Trans. Inf. Theory.

[11]  Eric M. Rains,et al.  Shadow Bounds for Self-Dual Codes , 1998, IEEE Trans. Inf. Theory.

[12]  M. Harada,et al.  Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6 , 2002 .

[13]  A. Atkin,et al.  Modular Forms , 2017 .

[14]  J. H. Conway,et al.  A Note on Optimal Unimodular Lattices , 1998 .

[15]  Masaaki Harada,et al.  Self-Dual Codes over Z 4 and Unimodular Lattices : A Survey , 2002 .

[16]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[17]  Christine Bachoc,et al.  Type II codes over Z4 , 1997, IEEE Trans. Inf. Theory.

[18]  N. Sloane,et al.  The Shadow Theory of Modular and Unimodular Lattices , 1998, math/0207294.

[19]  N. J. A. Sloane,et al.  An Upper Bound for Self-Dual Codes , 1973, Inf. Control..