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33
Type II Codes, Even Unimodular Lattices and Invariant Rings
, 1999
"... In this paper, we study selfdual codes over the ring Z 2k of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of This work was supported in part by a grant from the Japan Society for the Promotion of Science. finite groups. We introduce T ..."
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Cited by 34 (10 self)
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In this paper, we study selfdual codes over the ring Z 2k of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of This work was supported in part by a grant from the Japan Society for the Promotion of Science. finite groups. We introduce Type II codes over Z 2k which are closely related to even unimodular lattices, as a remarkable class of selfdual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z 6 of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z 2k are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z 2k and we give the Molien se...
Complementary Sets, Generalized ReedMuller Codes, and Power Control for OFDM
 IEEE Trans. Inform. Theory
, 2007
"... The use of errorcorrecting codes for tight control of the peaktomean envelope power ratio (PMEPR) in orthogonal frequencydivision multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each qphase (q is even) sequence of len ..."
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The use of errorcorrecting codes for tight control of the peaktomean envelope power ratio (PMEPR) in orthogonal frequencydivision multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each qphase (q is even) sequence of length 2 m lies in a complementary set of size 2 k+1, where k is a nonnegative integer that can be easily determined from the generalized Boolean function associated with the sequence. For small k this result provides a reasonably tight bound for the PMEPR of qphase sequences of length 2m. A new 2hary generalization of the classical Reed–Muller code is then used together with the result on complementary sets to derive flexible OFDM coding schemes with low PMEPR. These codes include the codes developed by Davis and Jedwab as a special case. In certain situations the codes in the present correspondence are similar to Paterson’s code constructions and often outperform them.
Low Density Parity Check Codes Over Groups and Rings
 In Proc. 2002 IEEE Information Theory Workshop
, 2002
"... The role of low density parity check principles in the design of group codes for coded modulation is examined. In this context, the structure of linear codes over certain rings Zm and Gm is discussed, and LDPC codes over these ring structures are designed. ..."
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Cited by 8 (0 self)
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The role of low density parity check principles in the design of group codes for coded modulation is examined. In this context, the structure of linear codes over certain rings Zm and Gm is discussed, and LDPC codes over these ring structures are designed.
Double Circulant Codes over Z4 and Even Unimodular Lattices
, 1997
"... With the help of some new results about weight enumerators of selfdual codes over Z4 we investigate a class of double circulant codes over Z4, one of which leads to an extremal even unimodular 40dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech latt ..."
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Cited by 8 (0 self)
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With the help of some new results about weight enumerators of selfdual codes over Z4 we investigate a class of double circulant codes over Z4, one of which leads to an extremal even unimodular 40dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech lattice”.
Linear codes over finite chain rings
 ELECTRONIC JOURNAL OF COMBINATORICS
, 1998
"... The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a wellknown result for linear codes over fields, we prove that there exists a onetoone correspondence between socalled fat linear codes over chain rings and multisets of ..."
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Cited by 8 (0 self)
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The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a wellknown result for linear codes over fields, we prove that there exists a onetoone correspondence between socalled fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings.
Construction of a (64, 2 , 12) Code via Galois Rings
, 1997
"... Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4 16. At length 16 they coincide to give the NordstromRobinson code. This paper constructs a nonlinear (64, 2 , 12) ..."
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Cited by 2 (0 self)
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Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4 16. At length 16 they coincide to give the NordstromRobinson code. This paper constructs a nonlinear (64, 2 , 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The NordstromRobinson code is defined in this same way, and like the NordstromRobinson code, the new code is better than any linear code that is presently known. Keywords: Algebraic coding theory, codes over rings 1.
Minimal trellis construction for finite support convolutional ring codes
 Coding Theory and Applications (ICMCTA), LN in Computer Science 5228
, 2008
"... Abstract. We address the concept of “minimal polynomial encoder ” for finite support linear convolutional codes over Zpr. These codes can be interpreted as polynomial modules which enables us to apply results from the 2007paper [8] to introduce the notions of “pencoder ” and “minimal pencoder”. H ..."
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Abstract. We address the concept of “minimal polynomial encoder ” for finite support linear convolutional codes over Zpr. These codes can be interpreted as polynomial modules which enables us to apply results from the 2007paper [8] to introduce the notions of “pencoder ” and “minimal pencoder”. Here the latter notion is the ring analogon of a row reduced polynomial encoder from the field case. We show how to construct a minimal trellis representation of a delayfree finite support convolutional code from a minimal pencoder. We express its number of trellis states in terms of a degree invariant of the code. The latter expression generalizes the wellknown expression in terms of the degree of a delayfree finite support convolutional code over a field to the ring case. The results are also applicable to block trellis realization of polynomial block codes over r, such as CRC codes over Zpr.
GENERATOR POLYNOMIALS OF THE pADIC QUADRATIC RESIDUE CODES
"... Abstract. Using the Newton’s identities, we give the inductive formula for the generator polynomials of the padic quadratic residue codes. 1. ..."
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Abstract. Using the Newton’s identities, we give the inductive formula for the generator polynomials of the padic quadratic residue codes. 1.
Four applications of Z4codes
 In Fifth Conference on Discrete Mathematics and Computer Science (Spanish
, 2006
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