arXiv:2501.17010 · Cryptography and Communications

New quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes

Oisin Campion, Fernando Hernando, Gary McGuire

Published Aug 13, 2025• 3 citations

Abstract

Abstract Let q be a prime power. Let $$\lambda >1$$ λ > 1 be a divisor of $$q-1$$ q - 1 and $$\tau >1$$ τ > 1 be a divisor of $$q+1$$ q + 1 . Under certain conditions we prove that there exists an MDS stabilizer quantum code with length $$n=\lambda \tau \sigma $$ n = λ τ σ for a range of values of $$\sigma $$ σ . This is a flexible construction, which includes previously known families and also new MDS parameters not known before.

DOI: 10.1007/s12095-025-00834-2

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arXiv:2501.17010 · Cryptography and Communications

New quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes

Oisin Campion, Fernando Hernando, Gary McGuire

Published Aug 13, 2025• 3 citations

arXiv HTML PDF

Abstract

Abstract Let q be a prime power. Let

\lambda >1

λ > 1 be a divisor of

q-1

q - 1 and

\tau >1

τ > 1 be a divisor of

q+1

q + 1 . Under certain conditions we prove that there exists an MDS stabilizer quantum code with length

n=\lambda \tau \sigma

n = λ τ σ for a range of values of

\sigma

σ . This is a flexible construction, which includes previously known families and also new MDS parameters not known before.

DOI: 10.1007/s12095-025-00834-2

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Abstract Let q be a prime power. Let

\lambda >1

λ > 1 be a divisor of

q-1

q - 1 and

\tau >1

τ > 1 be a divisor of

q+1

q + 1 . Under certain conditions we prove that there exists an MDS stabilizer quantum code with length

n=\lambda \tau \sigma

n = λ τ σ for a range of values of

\sigma

σ . This is a flexible construction, which includes previously known families and also new MDS parameters not known before.

q

-ary quantum MDS codes with minimum distance larger than

q/2+1

. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.

Quantum MDS Codes with length

n\equiv 0,1(

mod

\,\frac{q\pm1}{2})

n

of most of the quantum MDS codes we constructed satisfies

n\equiv 0,1(

mod

\,\frac{q\pm1}{2})

, which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have large minimum distances that are greater than

q/2+1

[[N,N-2K,K+1]]_q

under the following conditions: i)

N=t(q-1)+1

t(q-1)+2

with

t|(q+1)

and

K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1

; ii)

(n-1)|(q^2-1)

N=n

N=n+1

K_0=\lfloor\frac{n+q-1}{q+1}\rfloor

, and

K\ge K_0+1

; iii)

N=tq+1

\forall~1\le t\le q

and

K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1

; iv)

n|(q^2-1)

n_2=\frac{n}{\gcd (n,q+1)}

\forall~ 1\le t\le \frac{q-1}{n_2}-1

N=(t+1)n+2

and

K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1

[N, N-2 K, K+1]_{q}

. We give conditions on the length N and the values of K of the dimension N-2K. Parameter comparisons over small fields highlight the novelty of the quantum codes that we obtain.