Optimal self-dual codes over F2×F2 with respect to the Hamming weight

Dept. of Bus. Inf. Sci., Jobu Univ., Gunma, Japan
IEEE Transactions on Information Theory (Impact Factor: 2.65). 03/2004; DOI: 10.1109/TIT.2003.822576
Source: IEEE Xplore

ABSTRACT In this paper, we study optimal self-dual codes and type IV self-dual codes over the ring F2×F2 of order 4. We give improved upper bounds on minimum Hamming and Lee weights for such codes. Using the bounds, we determine the highest minimum Hamming and Lee weights for such codes of lengths up to 30. We also construct optimal self-dual codes and type IV self-dual codes.

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    ABSTRACT: Gleason’s theorem on self-dual binary codes says that the associated space of weight enumerators coincides with the invariant algebra of a certain finite complex matrix group. The main ingredient here is the MacWilliams identity relating the weight enumerators of a code and its dual. For doubly-even self-dual binary codes the relevant group is an exceptional two-dimensional unitary reflection group of order 192, implying that the invariant algebra is polynomial, being generated in degrees 8 and 24 by the weight enumerators of the extended Hamming and Golay codes of the respective lengths. Motivated by this and similar results for various other types of codes, in the book under review the general notions of form rings and their representations are introduced. The ingredients essentially are the ring R over which the codes under consideration are written, and the sets of bilinear forms M and quadratic forms Φ with values in a suitable abelian group A, responsible for the notions of duality and isotropy, respectively, together with the relations between M and Φ. A representation space V of a form ring now gives rise to a type of codes, which are the R-submodules of V ⊕n , for all n∈ℕ. If R is finite and A=ℚ/ℤ, the natural action of the unit group R * on the group algebra ℂ[V] extends to R * ⋉Φ, where φ∈Φ acts by φ:v↦exp(2πi·φ(v))·v. A code C≤V has an obvious enumerator in ℂ[V], which is invariant under R * ⋉Φ if and only if C is isotropic. If C moreover is self-dual its enumerator also is invariant under certain generalised MacWilliams transforms, where if R is a field these are just the maps v↦(1/|V|)∑ w∈V exp(2πi·β(w,v))·w, for β∈M, classically known from the MacWilliams identity. The group generated by R * ⋉Φ and the generalised MacWilliams transforms is finite, and called the associated Clifford-Weil group ℭ(V). For example, for doubly-even self-dual binary codes, the field R:=𝔽 2 is given an appropriate form ring structure, and the Clifford-Weil group associated to the natural R-module V:=𝔽 2 is a two-dimensional complex matrix group coinciding with the group from Gleason’s theorem. The main results of this book now say that for a broad class of finite rings, including all finite fields, the space of ℭ(V)-invariants in ℂ[V] indeed is spanned by the enumerators of self-dual isotropic codes in V. This implies that the space of homogeneous ℭ(V)-invariants of degree n in the symmetric algebra S(ℂ[V]) is spanned by the complete weight enumerators of self-dual isotropic codes in V ⊕n , which is a far-reaching generalisation of Gleason’s theorem. Here is a short account of the contents of the various chapters: Chapters 1, 3, 4, and 5 provide the necessary theoretical framework culminating in the proof of the main results just mentioned. In particular, in doing so, a Morita theory for form rings in an abstract categorical setting is developed. Chapters 2, 6, 7, and 8 contain comprehensive lists of examples, in particular covering all types of classical self-dual codes over finite fields, such as binary, Euclidean and Hermitian codes, as well as non-classical types, such as codes over ℤ/4ℤ. The associated Clifford-Weil groups are determined, and for codes over small rings such as 𝔽 2 , 𝔽 3 , 𝔽 4 , where the Clifford-Weil groups typically are closely related to unitary reflection groups, their Molien series and where possible explicit generators of their invariant algebras are given. Chapter 10 as an application gives a description of the complete weight enumerators of maximal isotropic codes of certain types. Chapters 11 and 12 collect the known data on extremal and optimal codes, and on the explicit enumeration of self-dual codes, which has been accumulated in the past decades. Finally, in the light of the new theory, Chapter 9 briefly elaborates on the well-known connection of codes and lattices, while Chapter 13 comments on quantum codes. This book, introducing a new unifying theory and its applications to a wealth of substantial examples, is certainly written for experts in the field. Although all the notions needed are introduced briefly, to understand the details of the theory and of the concise proofs, some background in coding theory, lattice theory, Witt groups, modular forms, general module theory, category theory, and quantum codes seems to be indispensable. Finally, it should be stressed that massive computations have been necessary to compile the explicit data presented, which, although the algorithmic and computational details are spared, justifies the appearance of this book in a series devoted to computational mathematics.
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    ABSTRACT: In this study, we consider linear and especially cyclic codes over the non-chain ring Zp[v]/⟨vp − v⟩ where p is a prime. This is a generalization of the case p = 3. Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance between codes over this ring and p-ary codes and moreover this map enlightens the structure of these codes. Furthermore, a MacWilliams type identity is presented together with some illustrative examples.
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    ABSTRACT: In this paper quadratic residue codes over the ring 𝔽 p +v𝔽 p are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes enjoy similar properties as quadratic residue codes over finite fields. For the case p=2, Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p-ary codes are obtained as images of quadratic residue codes over 𝔽 p +v𝔽 p in the cases where p is an odd prime. These results are presented in tables.
    Journal of Pure and Applied Algebra 11/2014; 11(11). DOI:10.1016/j.jpaa.2014.03.002 · 0.58 Impact Factor


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