scholarly journals Construction of Barnes-Wall lattices from linear codes over rings

2012 ◽  
Author(s):  
J. Harshan ◽  
Emanuele Viterbo ◽  
J.-C. Belfiore
Keyword(s):  
Linear Codes ◽  
Codes Over Rings ◽  
Linear Codes Over Rings

Quantum codes over rings

2014 ◽  
Vol 12 (04) ◽  
pp. 1450020 ◽  
Author(s):  
Kenza Guenda ◽  
T. Aaron Gulliver
Keyword(s):  
Linear Codes ◽  
Error Correcting Codes ◽  
Codes Over Rings ◽  
Quantum Codes ◽  
Matrix Product ◽  
Frobenius Rings ◽  
Linear Codes Over Rings ◽  
Generalized Gray Map ◽  
Css Construction ◽  
Quantum Error

This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽pk are also obtained from linear codes over rings using the generalized Gray map.

scholarly journals Construction of Linear Codes over Rings Zm Correcting Double ±1 or ±2 Errors

2018 ◽  
pp. 66-73
Author(s):  
Gurgen Khachatrian ◽  
Hamlet Khachatrian
Keyword(s):  
Linear Codes ◽  
Codes Over Rings ◽  
Optimal Codes ◽  
Absolute Value ◽  
Linear Codes Over Rings

In this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings Z5, Z7 and Z9 is presented with the limitation that both errors have the same amplitude in absolute value.

On modules with cyclic socle

2016 ◽  
Vol 15 (09) ◽  
pp. 1650162 ◽  
Author(s):  
Ali Assem
Keyword(s):  
Coding Theory ◽  
Linear Codes ◽  
Extension Theorem ◽  
Extension Property ◽  
Codes Over Rings ◽  
Extension Problem ◽  
Hamming Weight ◽  
Frobenius Rings ◽  
Code Equivalence ◽  
Open Question

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

Two-Weight Linear Codes and Their Applications in Secret Sharing

2019 ◽  
Vol 28 (4) ◽  
pp. 706-711
Author(s):  
Yaru Wang ◽  
Fulin Li ◽  
Shixin Zhu
Keyword(s):  
Secret Sharing ◽  
Linear Codes
Sign in / Sign up

Export Citation Format

Share Document