Quantum codes over rings

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.

[1]  Marcus Greferath,et al.  Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15) code , 1999, IEEE Trans. Inf. Theory.

[2]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[3]  Jay A. Wood.  Duality for modules over finite rings and applications to coding theory , 1999 .

[4]  Vera Pless,et al.  Cyclic codes and quadratic residue codes over Z4 , 1996, IEEE Trans. Inf. Theory.

[5]  Bram van Asch,et al.  Matrix-product codes over finite chain rings , 2008, Applicable Algebra in Engineering, Communication and Computing.

[6]  Xiang-dong Hou.  Finite Frobenius Rings and Their Applications , 2007 .

[7]  Jose Maria P. Balmaceda,et al.  Mass formula for self-dual codes over Zp2 , 2008, Discret. Math..

[8]  N. J. A. Sloane,et al.  Self-Dual Codes over the Integers Modulo 4 , 1993, J. Comb. Theory, Ser. A.

[9]  Graham H. Norton,et al.  Matrix-Product Codes over ?q , 2001, Applicable Algebra in Engineering, Communication and Computing.

[10]  Masaaki Harada,et al.  Type II Codes Over F2 + u F2 , 1999, IEEE Trans. Inf. Theory.

[11]  T. Honold,et al.  Weighted modules and representations of codes , 1998 .

[12]  Santosh Kumar,et al.  Nonbinary Stabilizer Codes Over Finite Fields , 2005, IEEE Transactions on Information Theory.

[13]  Andreas Klappenecker,et al.  Stabilizer codes over Frobenius rings , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[14]  Chaoping Xing,et al.  Asymptotic bounds on quantum codes from algebraic geometry codes , 2006, IEEE Transactions on Information Theory.

[15]  Lih-Yuan Deng,et al.  Orthogonal Arrays: Theory and Applications , 1999, Technometrics.

[16]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[17]  Jon-Lark Kim,et al.  MDS codes over finite principal ideal rings , 2009, Des. Codes Cryptogr..

[18]  Alfred Wassermann,et al.  Minimum Weights and Weight Enumerators of $\BBZ_{4}$-Linear Quadratic Residue Codes , 2012, IEEE Transactions on Information Theory.