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Orphan structure of the first-order Reed-Muller codes

Richard A. Brualdi, Ning Cai, Vera S. Pless
1992 Discrete Mathematics  
Cai and VS. Pless, Orphan structure of the first-order Reed-Muller codes, Discrete Mathematics 102 (1991) 239-247. We investigate a method of combining two codes which we call the outer product.  ...  First-order Reed-Muller codes are outer products of a number of copies of the full binary space of length 2, and we apply our results to obtain cosets of the Reed-Muller codes which have no ancestors,  ...  The first-order Reed-Muller code R(1, m) is the outer product of m first order Reed-Muller codes R(1, 1).  ... 
doi:10.1016/0012-365x(92)90118-y fatcat:ldwuv7uotbfltgmqq2f73mmmg4

Covering Radius of the Reed–Muller CodeR(1, 7)—A Simpler Proof

Xiang-dong Hou
1996 Journal of combinatorial theory. Series A  
A deep result about the Reed Muller codes, proved by Mykkeltveit in 1980, is that the covering radius of the Reed Muller code R(1, 7) equals 56.  ...  Among the results about \(1, m), that \(1, 7)=56, proved by Mykkeltveit [9], is one of the most difficult.  ...  Introduction Let R(r, m) be the rth order Reed Muller code of length 2 m , and let \(r, m) be its covering radius.  ... 
doi:10.1006/jcta.1996.0055 fatcat:aphetd6usbay7dkzi52mfsrk3i

Page 1252 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews  
We characterize cosets that are orphans, and then prove the existence of a family of orphans of first order Reed-Muller codes R(1,m).  ...  Richard A. (1-WI); Pless, Vera S. (1-ILCC) Orphans of the first order Reed-Muller codes.  ... 

Classification of cosets of the Reed-Muller code R(m−3,m)

Xiang-dong Hou
1994 Discrete Mathematics  
Cosets of the Reed-Muller code R(m-3,tn) are classified under the actions of GL(m,2) and GA(m, 2), the latter being the automorphism group of R(m -3, m) for m > 4.  ...  The number of cosets in each class is calculated. Orphans of R(m-3, m) are identified, and the normality of R(m-3, m) is established.  ...  G-equivalent cosets have the same coding theoretic properties. In this paper, we concentrate on the cosets of the (m -3)rd order Reed-Muller code R( m -3, m) of length 2".  ... 
doi:10.1016/0012-365x(94)90113-9 fatcat:5rkol7flgfgibg7yfabbzxao7a

On some cosets of the first-order Reed-Muller code with high minimum weight

C. Fontaine
1999 IEEE Transactions on Information Theory  
We study a family of particular cosets of the first-order Reed-Muller code R(1; m): those generated by special codewords, the idempotents.  ...  Thus we obtain new maximal weight distributions of cosets of R(1; 7) and 84 distinct almost maximal weight distributions of cosets of R(1; 9), that is, with minimum weight 240.  ...  Charpin and C. Carlet for their motivating discussions and N. Sendrier for valuable improvements concerning the implementation. We wish to thank H. F. Mattson and E. F.  ... 
doi:10.1109/18.761276 fatcat:rfor5ftkqjhb5i5iteq3ukmhl4

Page 3115 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
cosets of the Reed-Muller code R(m — 3, m).  ...  The covering radius is defined as the weight of the coset of greatest weight; it is the maximum distance of any vector in the space from the code. In a paper on RM, G. Seroussi and A.  ... 

Page 1032 of Mathematical Reviews Vol. , Issue 94b [page]

1994 Mathematical Reviews  
II (224-233); Philippe Langevin, On the orphans and covering radius of the Reed-Muller codes (234-240); Yuan Xing Li and Xin Mei Wang, A joint authentication and encryption scheme based on algebraic coding  ...  Janwa, On the parameters of algebraic geometric codes (19-28); Erich Kaltofen and B. David Saunders, On Wiedemann’s method of solving sparse linear systems (29-38); Simon N. Litsyn [S. N.  ... 

Page 1915 of Mathematical Reviews Vol. 26, Issue Index [page]

Mathematical Reviews  
(English summary) 94b:94033 Langevin, Philippe On the orphans and covering radius of the Reed-Muller codes. (English summary) (see 94b:68002) Litsyn, S.  ...  ., 94m:94030 Hou, Xiang Dong Some results on the covering radii of Reed-Muller codes. (English summary) 94f:94020 Katsman, G. L. Bounds on covering radius of dual product codes.  ... 

Page 1770 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
Orphans of the first order Reed-Muller codes. 91b:94044 Bruck, Jehoshua (with Naor, Moni) The hardness of decoding linear codes with preprocess- ing. (Not in MR) Bussemaker, F.C.  ...  A new table of constant weight codes. 91h:94028 Brualdi, Richard A. (with Pless, Vera S.) On the length of codes with a given covering radius. 91¢:94025 — (with Pless, Vera S.)  ... 

Page 1835 of Mathematical Reviews Vol. 25, Issue Index [page]

Mathematical Reviews  
On the covering radius of Reed-Muller codes. (English summary) 93¢e:94016 Conway, J. H. (with Pless, Vera S.; Sloane, N. J. A.)  ...  Orphan structure of the first-order Reed-Muller codes. 93b:94021 Cai, Ning see Brualdi, Richard A.; et al., 93b:94021 Calderbank, A. R. Covering machines.  ... 

Covering radius 1985-1994

G.D. Cohen, S. Litsyn, A. Lobstein, H.F. Mattson
Proceedings of 1995 IEEE International Symposium on Information Theory  
The proofs of Propositions 2.5 and 2.6, by induction on t, are based on the fact that codes with length n, covering radius t, and containing K(n, t) words (i.e., optimal covering codes) cannot be too unbalanced  ...  Table A gives bounds for K(The so-called sphere-covering bound states that if C is a code of length n and covering radius t, then the volume of a sphere of radius t, multiplied by the cardinality of C,  ...  Specific Classes of Codes Covering Radius of Reed-Muller Codes Reed-Muller (RM) codes are among the most interesting families in the study of covering radius.  ... 
doi:10.1109/isit.1995.535749 fatcat:vqartrp3offzdj6ols6lnqeht4

Page 150 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(From the text) 91a:94027 94B05 — (with Pless, Vera S.) Orphans of the first order Reed-Muller codes. JEEE Trans. Inform. Theory % (1990), no. 2, 399-401.  ...  On the length of codes with a given covering radius. Coding theory and design theory, Part I, 9-15, IMA Vol. Math. Appl., 20, Springer, New York, 1990.  ... 

Page 978 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
On the length of codes with a given covering radius. Coding theory and design theory, Part I, 9-15, IMA Vol. Math. Appl., 20, Springer, New York, 1990.  ...  (From the text) 91a:94027 94B05 — (with Brualdi, Richard A.) Orphans of the first order Reed-Muller codes. JEEE Trans. Inform. Theory % (1990), no. 2, 399-401.  ... 

Boolean Functions for Cryptography and Error-Correcting Codes [chapter]

Claude Carlet, Yves Crama, Peter L. Hammer
Boolean Models and Methods in Mathematics, Computer Science, and Engineering  
Acknowledgement We thank Caroline Fontaine for her careful reading of a previous draft of this chapter.  ...  of the covering radius of the Reed-Muller code of order 1 if n is even; indeed, in the case of the Reed-Muller code of order 1, the covering radius coincides with the maximum nonlinearity of Boolean functions  ...  (the nonlinearity of f ) equals 2 n−1 − 2 n/2−1 (the covering radius of the Reed-Muller code of order 1).  ... 
doi:10.1017/cbo9780511780448.011 fatcat:dtgopxbkmjgahepxw46xgo6rd4

Page 732 of Mathematical Reviews Vol. 26, Issue Index [page]

Mathematical Reviews  
Langevin, Philippe On the orphans and covering radius of the Reed-Muller codes.  ...  (see 94i:00015) 65M06 (76M25) Langsetmo, Lisa The K-theory localization of loops on an odd sphere and applications. Topology 32 (1993), no. 3, 577-585.  ... 
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