On the hulls of cyclic codes of oddly even length over [formula omitted

Abstract

In this paper, we study hulls of cyclic codes of length 2n (i.e., oddly even length) over the ring Z4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z4[x]/<x2n−1>, where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2n over Z4 in terms of the generator polynomials of the code C, and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0≤Θ≤2n. We also obtain an enumeration formula for cyclic codes of length 2n over Z4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E(2n), of the hulls of cyclic codes of length 2n over Z4 and establish a formula for E(2n) that handles well. With the help of this formula, we deduce that E(2n)=5n6 when n∈N2, and we study the growth rate of E(2n) with respect to the length 2n when n∉N2, where N2 denotes the set of all positive integers ω such that w divides 2i+1 for some positive integer i. Further, when n∉N2, we derive lower and upper bounds on E(2n) and show that these bounds are attained at n=7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z4 with specific parameters from cyclic codes of oddly even length over Z4.