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Minimum weight bases for quaternary Reed–Muller codes

F. I. Solov'eva
2021 Sibirskie Elektronnye Matematicheskie Izvestiya  
We have found one more general property for the families of quaternary Reed Muller codes that is common with binary Reed Muller codes: all these quaternary codes have bases of minimum weight codewords.  ...  The quaternary Plotkin and BQ-Plotkin constructions giving the families of quaternary Reed Muller codes were presented in 2009.  ...  In the paper we investigate such property for the families of quaternary linear Reed Muller codes from [10, 11] .  ... 
doi:10.33048/semi.2021.18.103 fatcat:impo2cq72bahxfxr6sxrg2wfpq

${\BBZ}_4$-Valued Quadratic Forms and Quaternary Sequence Families

Kai-Uwe Schmidt
2009 IEEE Transactions on Information Theory  
This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary sequence sets.  ...  Then, motivated by practical considerations, a subset S 3 (t) of family S(t) is defined, and the correlation distribution of family S 3 (t) is given for odd and even m.  ...  II.C], for example), the theory of -valued quadratic forms can be used to analyze the Lee weight distribution of subcodes of the quaternary second-order Reed-Muller code , such as the quaternary Kerdock  ... 
doi:10.1109/tit.2009.2032818 fatcat:7nx2ldr7nrfsdgkik3a2ffr6nm

Page 9449 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
For an #(r,m) code, this operation yields a linear quaternary code with the same length, dimension, and minimum distance as the Reed-Muller #(r — 1,m — 2) code.  ...  These lineav codes have parameters identical to those of binary Reed-Muller codes and some of their properties are also similar to those of RM codes.  ... 

Construction of Z4-linear Reed-Muller codes [article]

J. Pujol, J. Rifá, F. I. Solov'eva
2008 arXiv   pre-print
properties as the codes in the usual binary linear Reed-Muller family.  ...  Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental  ...  Quaternary Reed-Muller codes The usual binary linear RM family of codes is one of the oldest and interesting family of codes.  ... 
arXiv:0801.3024v1 fatcat:lpcd4jj7zbcmvmbpld7i7rm2dq

On the apparent duality of the Kerdock and Preparata codes [chapter]

A. Roger Hammons, P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, Patrick Solé
1993 Lecture Notes in Computer Science  
The quaternary code Q is the Z Z4-analog of the rst-order Reed-Muller code. As a result, PL has a simple description in the Z Z4-domain that admits a simple syndrome decoder.  ...  The construction involves quaternary codes over the ring Z Z4 of integers modulo 4. We exhibit a quaternary code Q and its quaternary dual Q ?  ...  Kerdock and Preparata codes, as well as results concerning the existence of quaternary versions of Reed-Muller, Golay and Hamming codes.  ... 
doi:10.1007/3-540-56686-4_30 fatcat:px5ghwocsfgkviqodgyhkkd67q

Quaternary Constant-Amplitude Codes for Multicode CDMA [article]

Kai-Uwe Schmidt
2009 arXiv   pre-print
families of quaternary constant-amplitude codes.  ...  Several constructions of such functions are presented, which are exploited in connection with algebraic codes over Z_4 (in particular quaternary Reed-Muller, Kerdock, and Delsarte-Goethals codes) to construct  ...  CODES FROM QUATERNARY REED-MULLER CODES Below we recall two quaternary generalizations of the binary Reed-Muller code from [5] .  ... 
arXiv:cs/0611162v2 fatcat:nob44epplbbq5flq4dod62idgu

Sequences for OFDM and Multi-Code CDMA: Two Problems in Algebraic Coding Theory [chapter]

Kenneth G. Paterson
2002 Sequences and their Applications  
Davis and the anonymous referee for comments which signi cantly improved the paper.  ...  W e h a ve also seen new motivation for attacking some well-known (and di cult) problems on Reed-Muller codes and bent functions.  ...  time model for an MC-CDMA system from Sect. etween c and the rst-order Reed-Muller code of length 2 m .  ... 
doi:10.1007/978-1-4471-0673-9_4 dblp:conf/seta/Paterson01 fatcat:q4suka3emndkvbgck2os2nptxe

Generalized Reed-Muller codes and power control in OFDM modulation

K.G. Paterson
2000 IEEE Transactions on Information Theory  
As a second consequence, the main result directly yields bounds on the PMEPRs of codes formed from selected cosets of the generalised rst order Reed-Muller code.  ...  In this paper, a p o w erful theory linking Golay complementary sets of polyphase sequences and Reed-Muller codes is developed.  ...  I am particularly grateful to Jonathan for many hours of helpful discussion.  ... 
doi:10.1109/18.817512 fatcat:eqelkni23zamhp2kv35srtpqii

Nonlinear codes defined by quadratic forms over GF(2)

Jean-Marie Goethals
1976 Information and Control  
An investigation is made of the family of codes which are supercodes of the first-order and subcodes of the second-order Reed-Muller codes.  ...  Finally, in Section 6, we show how easily the weight enumerators of some cyclic subcodes of the second-order Reed-Muller codes can be derived by our methods.  ...  In this paper, we shall mainly be concerned with the family of codes that are supercodes of the first-order and subcodes of the second-order Reed-Muller codes.  ... 
doi:10.1016/s0019-9958(76)90384-3 fatcat:iu5jwneiangh7nqubeypn2tjam

Z2Z4-linear codes: rank and kernel [article]

Cristina Fernandez-Cordoba, Jaume Pujol, Merce Villanueva
2009 arXiv   pre-print
For each possible rank r between these bounds, the construction of a Z2Z4-linear code with rank r is given.  ...  code (respectively, a quaternary linear code).  ...  Note that the rank of these two families of Z 2 Z 4 -linear codes satisfies the upper bound. Example 3: Let QRM (r, m) be the class of Z 4 -linear Reed-Muller codes defined in [3] .  ... 
arXiv:0807.4247v2 fatcat:jbfesqrawjeyjbbhy4l2zgriaq

On Codes With Low Peak-to-Average Power Ratio for Multicode CDMA

K.G. Paterson
2004 IEEE Transactions on Information Theory  
The problem of designing such codes is reformulated as a new coding-theoretic problem: codes with low PAPR are ones in which the codewords are far from the first-order Reed-Muller code.  ...  flexible parameters for the small values of n of practical interest.  ...  Lauder for his early interest in the work and for his help with the analysis in Section IV.  ... 
doi:10.1109/tit.2004.825505 fatcat:th6w5eathfe2lakw5fuxmtnfnq

On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification [chapter]

Denis S. Krotov, Mercè Villanueva
2015 Coding Theory and Applications  
It is known that there are exactly t−1 2 and t 2 nonequivalent Z 2 Z 4 -linear Hadamard codes of length 2 t , with α = 0 and α = 0, respectively, for all t ≥ 3.  ...  "On the automorphism groups of the Z₂Z₄-linear Hadamard codes and their classificati-  ...  Acknowledgements The work of the first author has been partially supported by the Russian Foundation for Basic Research under Grant 13-01-00463-a and by the Target Program of SB RAS for 2012-2014 (integration  ... 
doi:10.1007/978-3-319-17296-5_25 dblp:conf/icmcta/KrotovV14 fatcat:trdv2spk55hfhdlryslzyz47we

Generalized Hamming weights of nonlinear codes and the relation to the Z/sub 4/-linear representation

I. Reuven, Y. Be'ery
1999 IEEE Transactions on Information Theory  
These generalized weights are determined by the entropy/length profile of the code. We show that this definition characterizes the performance of nonlinear codes on the wire-tap channel of type II.  ...  We also show that the binary image of a Z 4 -linear code is a symmetric, not necessarily rectangular code.  ...  Cohen for directing them to [3] and [7] . The discussions with him while the second author visited ENST, Paris, were the origin of the new definition of the GHW. The authors also wish to thank Y.  ... 
doi:10.1109/18.749020 fatcat:c6ots4qjznarxh3et2cdhuwvmq

On the Secrecy Gain of Formally Unimodular Construction A_4 Lattices [article]

Maiara F. Bollauf, Hsuan-Yin Lin, Øyvind Ytrehus
2022 arXiv   pre-print
Furthermore, a new code construction of formally self-dual ℤ_4-linear codes is presented.  ...  Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted  ...  Definition 7 (Reed-Muller codes [19, Ch. 13]): For a given m ∈ N, the r-th order binary Reed-Muller code R(r, m) is a linear [n = 2 m , k = r i=0 m i ] code for r ∈ [0 : m], constructed as the vector  ... 
arXiv:2202.09236v1 fatcat:bvy7tmetxffkxj6jwrobfnwmym

A new binary sequence family with low correlation and large size

Nam Yul Yu, Guang Gong
2006 IEEE Transactions on Information Theory  
For odd  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments. The authors' research is supported by NSERC Grant RGPIN 227700-00.  ...  In Section III, we present new families is derived in terms of the weight distribution of a linear cyclic subcode of the second order Reed-Muller code.  ... 
doi:10.1109/tit.2006.871062 fatcat:btqb246qgnafdmjxbzv2g7v2u4
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