Relative Hulls and Quantum Codes

Given two q -ary codes C_{1} and C_{2} , the relative hull of C_{1} with respect to C_{2} is the intersection C_{1}\cap C_{2}^{\perp} . We prove that when q>2 , the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes wi...

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Bibliographic Details
Published in:IEEE transactions on information theory 2024-05, Vol.70 (5), p.3190-3201
Main Authors: Anderson, Sarah E., Camps-Moreno, Eduardo, Lopez, Hiram H., Matthews, Gretchen L., Ruano, Diego, Soprunov, Ivan
Format: Article
Language:English
Subjects:
Codes
CSS construction
entanglement-assisted quantum error-correcting codes
Equivalence
Error correcting codes
Error correction
Error correction codes
Hull
Hulls
Linear codes
quantum codes
Quantum entanglement
Upper bound
Vectors
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Summary:Given two q -ary codes C_{1} and C_{2} , the relative hull of C_{1} with respect to C_{2} is the intersection C_{1}\cap C_{2}^{\perp} . We prove that when q>2 , the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the e -Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when q>2 . We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2024.3373550