Author
F. J. MacWilliams
Bio: F. J. MacWilliams is an academic researcher from Bell Labs. The author has contributed to research in topics: Circulant matrix & Order (group theory). The author has an hindex of 4, co-authored 4 publications receiving 291 citations.
Papers
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TL;DR: C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥ are studied, and an upper bound obtained on the minimum distance is obtained.
172 citations
TL;DR: The object of this paper is to report that the number of codewords of weight 15 is zero, thus reducing the number-of-unknown weights to two.
81 citations
TL;DR: This paper provides methods (suitable for the computer) by which to construct orthogonal circulant matrices of any size, and finds the orders of these groups for n prime to p, and shows how to construct Gsp, Ssp, Osp from Gs, Ss, Os.
48 citations
TL;DR: This paper shows that Pelikan's conjecture is false for infinitely many values of n, and it is also shown that the conjecture is true for most values ofN, and a characterization is given of those values ofn for which it fails.
4 citations
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29 Jun 1997
TL;DR: In this article, the problem of finding quantum error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a trace inner product.
Abstract: The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment. In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
1,525 citations
TL;DR: It is shown that the minimal distance d of a binary self-dual code of length n>or=74 is at most 2((n+6)/10).
Abstract: It is shown that the minimal distance d of a binary self-dual code of length n>or=74 is at most 2((n+6)/10). This bound is a consequence of some new conditions on the weight enumerator of a self-dual code obtained by considering a particular translate of the code, called its shadow. These conditions also enable one to find the highest possible minimal distance of a self-dual code for all n>or=60; to show that self-dual codes with d or=22, with d>or=8 exist precisely for n=24, 32 and n>or=26, and with d>or=10 exist precisely for n>or=46; and to show that there are exactly eight self-dual codes of length 32 with d=8. Several of the self-dual codes of length 34 have trivial group (this appears to be the smallest length where this can happen). >
384 citations
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TL;DR: A survey of self-dual codes is given in this paper, where the authors present a comprehensive bibliography of codes over F2, F3, F4, Fq, Z4, Zm, shadow codes, weight enumerators, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes.
Abstract: A survey of self-dual codes, written for the Handbook of Coding Theory. Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this chapter include codes over F2, F3, F4, Fq, Z4, Zm, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.
329 citations
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AT&T1
TL;DR: In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product.
Abstract: The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
220 citations
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TL;DR: These methods are being developed to study the well known conjectures that every finite projective plane with no proper subplane is isomorphic to a prime field plane and that the order of a finite projectives plane is a power of a prime number.
Abstract: Semiadditive rings are defined and their relationship with the projective planes is studied. Free semiadditive rings provide an analogue of the ring of integers and polynomials for the ternary rings. A structure theory for free semiadditive rings is developed. It is shown that each element of a large class of semiadditive rings is obtained from a quotient of a polynomial ring over integers by an additive subgroup, by twisting addition and multiplication. This class includes all planar ternary rings. These methods are being developed to study the well known conjectures that every finite projective plane with no proper subplane is isomorphic to a prime field plane and that the order of a finite projective plane is a power of a prime number. Applications to these problems will be discussed in part II.
217 citations