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Orphan structure of the first-order Reed-Muller codes

1992
*
Discrete Mathematics
*

Cai

doi:10.1016/0012-365x(92)90118-y
fatcat:ldwuv7uotbfltgmqq2f73mmmg4
*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). ...##
###
Covering Radius of the Reed–Muller CodeR(1, 7)—A Simpler Proof

1996
*
Journal of combinatorial theory. Series A
*

A deep result about

doi:10.1006/jcta.1996.0055
fatcat:aphetd6usbay7dkzi52mfsrk3i
*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*. ...##
###
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*. ...##
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Classification of cosets of the Reed-Muller code R(m−3,m)

1994
*
Discrete Mathematics
*

Cosets

doi:10.1016/0012-365x(94)90113-9
fatcat:5rkol7flgfgibg7yfabbzxao7a
*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". ...##
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On some cosets of the first-order Reed-Muller code with high minimum weight

1999
*
IEEE Transactions on Information Theory
*

We study a family

doi:10.1109/18.761276
fatcat:rfor5ftkqjhb5i5iteq3ukmhl4
*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. ...##
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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. ...

##
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Covering radius 1985-1994

*
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*. ...

##
###
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. ...

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

*
Boolean Models and Methods in Mathematics, Computer Science, and Engineering
*

Acknowledgement We thank Caroline Fontaine for her careful reading

doi:10.1017/cbo9780511780448.011
fatcat:dtgopxbkmjgahepxw46xgo6rd4
*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). ...##
###
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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