TY - JOUR

T1 - On ZpZpk-additive codes and their duality

AU - Shi, Minjia

AU - Wu, Rongsheng

AU - Krotov, Denis S.

N1 - Funding Information: This work was supported in part by the National Natural Science Foundation of China under Grant 61672036, in part by the Excellent Youth Foundation of Natural Science Foundation of Anhui Province under Grant 1808085J20, and in part by the Russian Academy of Sciences, Siberian Branch, through the Program for Fundamental Scientific Studies under Grant 0314-2016-0016. Funding Information: Manuscript received August 23, 2018; revised October 27, 2018; accepted November 22, 2018. Date of publication November 30, 2018; date of current version May 20, 2019. This work was supported in part by the National Natural Science Foundation of China under Grant 61672036, in part by the Excellent Youth Foundation of Natural Science Foundation of Anhui Province under Grant 1808085J20, and in part by the Russian Academy of Sciences, Siberian Branch, through the Program for Fundamental Scientific Studies under Grant 0314-2016-0016. Publisher Copyright: © 1963-2012 IEEE. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - In this paper, two different Gray-like maps from Z p α × Z pk β , where p is prime, to Z p n , n=α +β p k-1 , denoted by φ and Φ, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if C is a Z p Z p k -additive code, and C⊥ is its dual, then the weight enumerators of the image p -ary codes φ (C) and Φ (C⊥) are formally dual. This is a partial generalization of [D. S. Krotov, On Z 2 k -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed Z p Z p 2 ⋯ Z p k alphabet is given.

AB - In this paper, two different Gray-like maps from Z p α × Z pk β , where p is prime, to Z p n , n=α +β p k-1 , denoted by φ and Φ, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if C is a Z p Z p k -additive code, and C⊥ is its dual, then the weight enumerators of the image p -ary codes φ (C) and Φ (C⊥) are formally dual. This is a partial generalization of [D. S. Krotov, On Z 2 k -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed Z p Z p 2 ⋯ Z p k alphabet is given.

KW - Additives

KW - Advanced Video Coding (AVC)

KW - Binary codes

KW - compression artifacts

KW - convolutional neural networks (CNN)

KW - Generators

KW - High Efficiency Video Coding (HEVC)

KW - Linear codes

KW - Measurement

KW - Propulsion

KW - video compression

KW - Zinc

KW - linear codes

KW - Gray map

KW - MacWilliams identity

KW - two-weight codes

KW - 1-perfect codes

KW - Dual codes

UR - http://www.scopus.com/inward/record.url?scp=85057805645&partnerID=8YFLogxK

U2 - 10.1109/TIT.2018.2883759

DO - 10.1109/TIT.2018.2883759

M3 - Article

AN - SCOPUS:85057805645

VL - 65

SP - 3841

EP - 3847

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 6

M1 - 8554305

ER -