On the hulls of cyclic codes of oddly even length over Z4

In this paper, we study hulls of cyclic codes of length 2n (i.e., oddly even length) over the ring Z(4) of integers modulo 4 by viewing these codes as ideals of the quotient ring Z(4)[x]/ < x(2n) - 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 Z(4) 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 Theta satisfying 0 <= Theta <= 2n. We also obtain an enumeration formula for cyclic codes of length 2n over Z(4) with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by epsilon(2n), of the hulls of cyclic codes of length 2n over Z(4) and establish a formula for epsilon(2n) that handles well. With the help of this formula, we deduce that epsilon(2n) = 5n/6 when n is not an element of N-2, and we study the growth rate of epsilon(2n) with respect to the length 2n when n is not an element of /N-2, where N-2 denotes the set of all positive integers omega such that w divides 2(i) +1 for some positive integer i. Further, when n is not an element of /N-2, we derive lower and upper bounds on epsilon(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 Z(4) with specific parameters from cyclic codes of oddly even length over Z(4).(c) 2023 Elsevier B.V. All rights reserved.