Improved Constructions of Nested Code Pairs

Abstract

Two new constructions of linear code pairs C 2 ⊂ C 1 are given for which the codimension and the relative minimum distances M 1 (C 1 , C 2 ), M 1 (C ⊥ 2 , C ⊥ 1 ) are good. By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes [40]. They can also be used to ensure reliable communication over asymmetric quantum channels [54]. The new constructions result from carefully applying the Feng-Rao bounds [21,31] to a family of codes defined from multivariate polynomials and Cartesian product point sets.

Two Gilbert–Varshamov-type existential bounds for asymmetric quantum error-correcting codes

“…Steane [16] first studied the asymmetry between probabilities of the bit and the phase errors, and he also considered QECC for asymmetric quantum errors, which are called asymmetric quantum error-correcting codes (AQECC). Research on AQECC has become very active recently, see [7,9,16] and the references therein.…”

“…Steane [16] first studied the asymmetry between probabilities of the bit and the phase errors, and he also considered QECC for asymmetric quantum errors, which are called asymmetric quantum error-correcting codes (AQECC). Research on AQECC has become very active recently, see [7,9,16] and the references therein.On the other hand, in the study of error-correcting codes, it is important to know the optimal performance of codes. For classical error-correcting codes, the Gilbert-Varshamov (GV) bound [11] is a sufficient condition for existence of codes whose parameters satisfies the GV bound.…”

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

“…Most of the asymmetric quantum codes come from the CSS construction of quantum stabilizer codes and, for them, there is also a Gilbert-Varshamov bound [35]. In addition, the existence of an asymmetric quantum error-correcting code coming from the CSS construction can also be applied to linear ramp secret sharing and communication over wiretap channels of type II [19].…”

New Binary and Ternary LCD Codes

“…Our LCD codes may be regarded as a generalization of BCH codes, including extensions to the case of more variables, and allow us to reach a wider variety of lengths. Their metric structure and duality properties have been studied and successfully used to construct quantum stabilizer codes in previous works of the authors [10,11,12,9,8].…”

“…Some good LCD codes in these families have the following parameters:[15,8,4] 2 , [153, 140, 4] 2 , [45, 32, 6] 2 , [135, 118, 6] 2 , [51, 40, ≥ 4] 2 and [153, 132, ≥ 6] 2 . All of them are optimal with the exception of the last two which are BKLC.…”