Two families of two-weight codes over $$\mathbb {Z}_4$$
TL;DR: Two infinite families of binary codes with two nonzero Lee weights with Gray images constructed by their generator matrices with the same weight distribution as the two-weight binary codes of type SU1.
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Abstract: Two infinite families of
$$\mathbb {Z}_4$$
-codes with two nonzero Lee weights are constructed by their generator matrices. Their Gray images are nonlinear with the same weight distribution as that of the two-weight binary codes of type SU1 in the sense of (Calderbank, Kantor, 1986).
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Citations
The Geometry of Two-Weight Codes Over ℤ p m
TL;DR: This result generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982) and relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry.
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Minimal and optimal binary codes obtained using $$C_D$$-construction over the non-unital ring I
Vidya Sagar,Ritumoni Sarma +1 more
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On Constructions of One-Lee Weight Codes Over Z₄
Zongbing Lin,Kaimin Cheng +1 more
TL;DR: In this paper, the authors constructed four infinite families of binary codes with one nonzero Lee weight by their generator matrices, and studied the linearity of their Gray images and obtained a family of optimal binary codes.
Certain binary minimal codes constructed using simplicial complexes
V.K. Sagar,Ritumoni Sarma +1 more
TL;DR: Sure, here is the TLDR: The manuscript studies binary minimal codes constructed using simplicial complexes over the non-chain ring $ \mathcal{R} = \frac{\mathbb{F}_2[u]}{\langle u^3 - u\rangle} $. Codes are constructed using the Delta sets and the ordered finite multiset $ D $. Conditions for minimality and self-orthogonality are obtained.
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References
The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes
TL;DR: Certain notorious nonlinear binary codes contain more codewords than any known linear code and can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes).
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•Posted Content
The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes
TL;DR: In this paper, it was shown that all the nonlinear binary codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals can be constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4.
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The Geometry of Two-Weight Codes
R. Calderbank,William M. Kantor +1 more
TL;DR: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles projectifs et certains graphes fortement reguliers as mentioned in this paper.
Weights of linear codes and strongly regular normed spaces
TL;DR: It is shown that every strongly regular normed space admits a representation by means of a projective code, which yields a one-to-one correspondence between two-weight projective codes over prime fields and some strongly regular graphs.
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Z4-linear Hadamard and extended perfect codes
TL;DR: In this article, it was shown that if N = 2 k ≥ 16, then there exist exactly ⌊( k − 1)/2⌋ pairwise nonequivalent Z 4 -linear Hadamard (N, 2 N, N /2)-codes and ⌈( k + 1)/ 2⌉ pairwise zero-rank extended perfect (N, 2 N/2 N, 4)-codes.
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