For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .

Defence Science JournalSep 2, 202115 citations

Construction of Quantum Codes from Constacyclic Codes Over the Class of Commutative Rings

Pushpendra Sharma, Amal S. Alali, Shakir Ali +1 more

This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.

Contemporary MathematicsAug 19, 20250 citations

On quantum and LCD Codes Over The Ring

F_q+vF_q+v^2F_q