Code2

Quaternary linear codes with one-dimensional Hermitian hull

Database for the paper “Shitao Li, Minjia Shi, Huizhou Liu, Optimal quaternary linear codes with one-dimensional Hermitian hull and related EAQECCs, Quantum Information Processing, 2023, 22(10): 388.”

Quaternary linear codes with one-dimensional Hermitian hull for $n≤12$.

n=2

[2, 1, 2] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1]

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n=3

[3, 1, 2] Linear Code over GF(2^2)
Generator matrix:
[1 1 0]

[3, 2, 1] Linear Code over GF(2^2)
Generator matrix:
[1 0 1]
[0 1 0]

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n=4

[4, 1, 4] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1]

[4, 2, 3] Quasicyclic of degree 2 Linear Code over GF(2^2)
Generator matrix:
[1 0 w w^2]
[0 1 w^2 w]

[4, 3, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1]
[0 1 0 1]
[0 0 1 1]

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n=5

[5, 1, 4] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 0]

[5, 2, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 w w^2 0]
[0 1 w^2 w 0]

[5, 3, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 0]
[0 1 0 1 0]
[0 0 1 1 0]

[5, 4, 1] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]

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n=6

[6, 1, 6] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1]

[6, 2, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 w^2 w 1 1]
[0 1 1 1 1 0]

[6, 3, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 w 0]
[0 1 0 w 1 1]
[0 0 1 1 1 1]

[6, 4, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w 1]
[0 1 0 0 1 1]
[0 0 1 0 1 0]
[0 0 0 1 1 0]

[6, 5, 2] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 1]
[0 1 0 0 0 1]
[0 0 1 0 0 1]
[0 0 0 1 0 1]
[0 0 0 0 1 1]

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n=7

[7, 1, 6] Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 0]

[7, 2, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 1 1 0 w w]
[0 1 1 1 1 1 1]

[7, 3, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w 0]
[0 1 0 w w 1 1]
[0 0 1 1 1 1 1]

[7, 4, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 0 1]
[0 1 0 0 w w 1]
[0 0 1 0 w 1 1]
[0 0 0 1 1 1 1]

[7, 5, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 1]
[0 1 0 0 0 0 1]
[0 0 1 0 0 0 1]
[0 0 0 1 0 0 1]
[0 0 0 0 1 0 1]

[7, 6, 1] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 1]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]

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n=8

[8, 1, 8] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 1 1]

[8, 2, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 1 0 0 w 1 1]
[0 1 1 1 1 1 1 1]

[8, 3, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w w^2 w w^2]
[0 1 0 w^2 w 1 0 w^2]
[0 0 1 w^2 1 1 w 1]

[8, 4, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w 1 1 1]
[0 1 0 0 1 w^2 w 0]
[0 0 1 0 w w 0 1]
[0 0 0 1 1 w^2 1 1]

[8, 5, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 1 w^2 0]
[0 1 0 0 0 0 1 w^2]
[0 0 1 0 0 w w 0]
[0 0 0 1 0 0 w w]
[0 0 0 0 1 1 w 1]

[8, 6, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 w^2 1]
[0 1 0 0 0 0 w^2 w]
[0 0 1 0 0 0 1 1]
[0 0 0 1 0 0 1 1]
[0 0 0 0 1 0 1 1]
[0 0 0 0 0 1 1 1]

[8, 7, 2] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1]
[0 0 1 0 0 0 0 1]
[0 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 1 1]

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n=9

[9, 1, 8] Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 1 1 0]

[9, 2, 7] Linear Code over GF(2^2)
Generator matrix:
[1 0 1 w w 1 0 1 w]
[0 1 w w 1 0 1 w w]

[9, 3, 6] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w^2 1 1]
[0 1 0 1 w^2 w^2 w 0 w]
[0 0 1 w w 0 w^2 w^2 1]

[9, 4, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w^2 1 0 1 w^2]
[0 1 0 0 w^2 1 1 w^2 0]
[0 0 1 0 0 1 1 1 1]
[0 0 0 1 1 0 1 w^2 w^2]

[9, 5, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 w^2 0 w^2 w]
[0 1 0 0 0 1 w^2 w 0]
[0 0 1 0 0 0 1 w^2 w]
[0 0 0 1 0 w w 0 1]
[0 0 0 0 1 1 w^2 1 1]

[9, 6, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 1 w^2 0]
[0 1 0 0 0 0 1 w 1]
[0 0 1 0 0 0 w w^2 1]
[0 0 0 1 0 0 w w 0]
[0 0 0 0 1 0 w^2 0 w^2]
[0 0 0 0 0 1 w^2 w^2 w]

[9, 7, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 1]

[9, 8, 1] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 1 1]

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n=10

[10, 1, 10] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1]

[10, 2, 7] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w w w 1 1 1]
[0 1 1 1 1 1 1 1 1 1]

[10, 3, 6] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w^2 0 w^2 w w 1]
[0 1 0 0 1 1 w w^2 1 1]
[0 0 1 0 1 w^2 w^2 w 0 w]

[10, 4, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w^2 1 0 1 w^2 0]
[0 1 0 0 w^2 1 1 w^2 0 0]
[0 0 1 0 0 1 1 1 1 0 0]
[0 0 0 1 1 0 1 w^2 w^2 0]

[10, 5, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 1 w w^2 0 w]
[0 1 0 0 0 w 1 w w w^2]
[0 0 1 0 0 w^2 1 1 w^2 0]
[0 0 0 1 0 0 1 1 1 1]
[0 0 0 0 1 1 0 1 w^2 w^2]

[10, 6, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 w^2 1 1 w^2]
[0 1 0 0 0 0 w^2 0 w^2 w]
[0 0 1 0 0 0 w 1 1 1]
[0 0 0 1 0 0 0 1 w^2 w]
[0 0 0 0 1 0 w w 0 1]
[0 0 0 0 0 1 1 w^2 1 1]

[10, 7, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 w^2 1 0]
[0 1 0 0 0 0 0 w 1 1]
[0 0 1 0 0 0 0 w 1 w^2]
[0 0 0 1 0 0 0 0 w w]
[0 0 0 0 1 0 0 w^2 0 1]
[0 0 0 0 0 1 0 0 w w^2]
[0 0 0 0 0 0 1 1 1 1]

[10, 8, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 w^2]
[0 1 0 0 0 0 0 0 w^2 1]
[0 0 1 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 1 1 1]

[10, 9, 2] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 1]

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n=11

[11, 1, 10] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 0]

[11, 2, 8] Linear Code over GF(2^2)
Generator matrix:
[1 0 1 0 1 w 1 1 0 w 1]
[0 1 0 1 w 1 1 w 1 1 1]

[11, 3, 7] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 1 w 1 w w 0]
[0 1 0 0 1 1 1 1 1 1 1]
[0 0 1 1 0 1 1 w w w^2 w^2]

[11, 4, 6] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w 0 w w 0 w 1 w]
[0 1 0 1 0 0 w^2 w w w^2 w^2]
[0 0 1 1 0 0 1 w^2 1 0 w^2]
[0 0 0 0 1 0 1 w w^2 w w^2]

[11, 5, 5] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 1 w w^2 0 w 0]
[0 1 0 0 0 w 1 w w w^2 0]
[0 0 1 0 0 w^2 1 1 w^2 0 0]
[0 0 0 1 0 0 1 1 1 1 0]
[0 0 0 0 1 1 0 1 w^2 w^2 0]

[11, 6, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 1 0 1 0 w]
[0 1 0 0 0 0 w 0 w 1 1]
[0 0 1 0 0 0 w 0 w^2 w^2 0]
[0 0 0 1 0 0 1 0 w^2 1 0]
[0 0 0 0 1 0 w^2 0 w w w^2]
[0 0 0 0 0 1 0 0 1 w^2 w]

[11, 7, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 w^2 1 1 w^2]
[0 1 0 0 0 0 0 w^2 0 w^2 w]
[0 0 1 0 0 0 0 w 1 1 1]
[0 0 0 1 0 0 0 1 w^2 w 0]
[0 0 0 0 1 0 0 0 1 w^2 w]
[0 0 0 0 0 1 0 w w 0 1]
[0 0 0 0 0 0 1 1 w^2 1 1]

[11, 8, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 w^2 1 0]
[0 1 0 0 0 0 0 0 w 1 1]
[0 0 1 0 0 0 0 0 w^2 1 w]
[0 0 0 1 0 0 0 0 w 1 w^2]
[0 0 0 0 1 0 0 0 0 w w]
[0 0 0 0 0 1 0 0 w^2 0 1]
[0 0 0 0 0 0 1 0 0 w w^2]
[0 0 0 0 0 0 0 1 1 1 1]

[11, 9, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 1]

[11, 10, 1] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 1 1]

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n=12

[12, 1, 12] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 1 1 1 1 1 1 1 1 1 1 1]

[12, 2, 9] Linear Code over GF(2^2)
Generator matrix:
[1 0 1 1 0 w w 1 0 1 1 1]
[0 1 1 1 1 1 1 0 1 1 w w^2]

[12, 3, 8] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 w w^2 w^2 1]
[0 1 0 w^2 w 1 w^2 1 1 w^2 1 w^2]
[0 0 1 w 1 1 1 1 1 0 0 w]

[12, 4, 7] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 w w^2 w^2 1]
[0 1 0 0 1 0 w^2 w w^2 1 0 w^2]
[0 0 1 0 w^2 w 1 w^2 0 1 w^2 w]
[0 0 0 1 1 w 0 1 w^2 w^2 w 0]

[12, 5, 6] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 1 w 0 w w^2 w^2 0]
[0 1 0 0 0 1 1 w 1 0 w 1]
[0 0 1 0 0 w w^2 w 1 1 w w]
[0 0 0 1 0 w 0 0 1 w^2 1 1]
[0 0 0 0 1 0 1 w 0 w^2 w w^2]

[12, 6, 6] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 1 w^2 1 1 w 1]
[0 1 0 0 0 0 w 0 1 w^2 w w^2]
[0 0 1 0 0 0 w w^2 w w^2 0 1]
[0 0 0 1 0 0 0 w w^2 w w^2 1]
[0 0 0 0 1 0 w^2 w 1 0 w^2 w]
[0 0 0 0 0 1 w^2 1 1 w 1 1]

[12, 7, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 w^2 1 1 w^2 0]
[0 1 0 0 0 0 0 w^2 0 w^2 w 0]
[0 0 1 0 0 0 0 w 1 1 1 0]
[0 0 0 1 0 0 0 1 w^2 w 0 0]
[0 0 0 0 1 0 0 0 1 w^2 w 0]
[0 0 0 0 0 1 0 w w 0 1 0]
[0 0 0 0 0 0 1 1 w^2 1 1 0]

[12, 8, 4] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 1 1 1 w]
[0 1 0 0 0 0 0 0 w w^2 0 w^2]
[0 0 1 0 0 0 0 0 w^2 1 1 w^2]
[0 0 0 1 0 0 0 0 w 1 1 1]
[0 0 0 0 1 0 0 0 1 w^2 w 0]
[0 0 0 0 0 1 0 0 0 1 w^2 w]
[0 0 0 0 0 0 1 0 w w 0 1]
[0 0 0 0 0 0 0 1 1 w^2 1 1]

[12, 9, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 w^2 w 0]
[0 1 0 0 0 0 0 0 0 w^2 0 1]
[0 0 1 0 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 0 0 w 1 1]
[0 0 0 0 1 0 0 0 0 w 1 w^2]
[0 0 0 0 0 1 0 0 0 w^2 1 1]
[0 0 0 0 0 0 1 0 0 0 1 w]
[0 0 0 0 0 0 0 1 0 1 w 1]
[0 0 0 0 0 0 0 0 1 0 w^2 w^2]

[12, 10, 2] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 w]
[0 1 0 0 0 0 0 0 0 0 w 1]
[0 0 1 0 0 0 0 0 0 0 1 1]
[0 0 0 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 0 0 0 1 1]
[0 0 0 0 0 0 1 0 0 0 1 1]
[0 0 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 1 1 1]

[12, 11, 2] Cyclic Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 1 1]

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Quaternary linear $[n,n-3]$ codes with one-dimensional Hermitian hull for $6≤n≤19$.

[6, 3, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w w w]
[0 1 0 1 w^2 w]
[0 0 1 w w 0]

[7, 4, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w w w^2]
[0 1 0 0 w^2 1 0]
[0 0 1 0 w^2 0 1]
[0 0 0 1 1 w^2 1]

[8, 5, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 w^2 1 w]
[0 1 0 0 0 0 w w]
[0 0 1 0 0 w^2 0 1]
[0 0 0 1 0 0 w w^2]
[0 0 0 0 1 1 1 1]

[9, 6, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 w^2 1]
[0 1 0 0 0 0 w 1 w^2]
[0 0 1 0 0 0 w^2 1 w]
[0 0 0 1 0 0 w w 0]
[0 0 0 0 1 0 1 w 1]
[0 0 0 0 0 1 w^2 w^2 1]

[12, 9, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 1 w^2]
[0 1 0 0 0 0 0 0 0 w w^2 w]
[0 0 1 0 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 0 0 w w^2 1]
[0 0 0 0 1 0 0 0 0 1 0 w]
[0 0 0 0 0 1 0 0 0 w w^2 0]
[0 0 0 0 0 0 1 0 0 0 w w]
[0 0 0 0 0 0 0 1 0 1 w^2 0]
[0 0 0 0 0 0 0 0 1 w w 0]

[13, 10, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 1 0 0 0 0 0 0 0 0 w^2 w^2 w]
[0 0 1 0 0 0 0 0 0 0 0 1 w^2]
[0 0 0 1 0 0 0 0 0 0 w w^2 w]
[0 0 0 0 1 0 0 0 0 0 1 0 w]
[0 0 0 0 0 1 0 0 0 0 1 w w]
[0 0 0 0 0 0 1 0 0 0 0 w w^2]
[0 0 0 0 0 0 0 1 0 0 1 w^2 0]
[0 0 0 0 0 0 0 0 1 0 w w 0]
[0 0 0 0 0 0 0 0 0 1 w 1 1]

[14, 11, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 0 0 1 1 w^2]
[0 0 1 0 0 0 0 0 0 0 0 w^2 w 1]
[0 0 0 1 0 0 0 0 0 0 0 0 1 w^2]
[0 0 0 0 1 0 0 0 0 0 0 w^2 1 1]
[0 0 0 0 0 1 0 0 0 0 0 1 w^2 w^2]
[0 0 0 0 0 0 1 0 0 0 0 w^2 w 0]
[0 0 0 0 0 0 0 1 0 0 0 w^2 1 w]
[0 0 0 0 0 0 0 0 1 0 0 0 w w]
[0 0 0 0 0 0 0 0 0 1 0 w 0 1]
[0 0 0 0 0 0 0 0 0 0 1 w^2 w^2 0]

[15, 12, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 w^2 w]
[0 1 0 0 0 0 0 0 0 0 0 0 1 w^2 1]
[0 0 1 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 0 0 1 0 0 0 0 0 0 0 0 w^2 w^2 w]
[0 0 0 0 1 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 1 w^2]
[0 0 0 0 0 0 1 0 0 0 0 0 w w^2 w]
[0 0 0 0 0 0 0 1 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 w w^2 1]
[0 0 0 0 0 0 0 0 0 1 0 0 1 w w]
[0 0 0 0 0 0 0 0 0 0 1 0 0 w w]
[0 0 0 0 0 0 0 0 0 0 0 1 w w 0]

[16, 13, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 w]
[0 1 0 0 0 0 0 0 0 0 0 0 0 1 w^2 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 w^2 w^2 w]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 w 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 w^2]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 w w]
[0 0 0 0 0 0 0 0 0 1 0 0 0 w w^2 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 w w]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 w w^2]
[0 0 0 0 0 0 0 0 0 0 0 0 1 w 1 1]

[17, 14, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 w]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 w 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 w w^2 w]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 w]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 w w^2 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 w w]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 w w^2]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 w^2 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 w w 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 w 1 1]

[18, 15, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 w]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 w^2 w^2 w]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 w^2]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 w w^2 w]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 w w^2 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 w]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 w w^2 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 w w]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 w w^2]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 w^2 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 w w 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w 1 1]

[19, 16, 3] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 w]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w^2 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 w^2 w^2 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 w^2 w^2 w]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 w 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 w^2]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 w w^2 w]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 w w^2 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 w w]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 w w^2 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 w w]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 w w^2]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 w^2 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 w 1 1]

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Quaternary linear $[n,3]$ codes with one-dimensional Hermitian hull for $n≥13$.

[13, 3, 9] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w 1 w w^2 0 w w^2 w^2 1]
[0 1 0 0 0 1 1 1 1 1 1 1 1]
[0 0 1 1 1 0 0 1 1 w w w^2 w^2]

[14, 3, 10] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 1 w w^2 w^2 w 1]
[0 1 0 w^2 w 1 w^2 1 0 1 w^2 1 w w^2]
[0 0 1 w 1 1 1 1 w 1 0 0 w^2 w]

[15, 3, 10] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 1 w w^2 w^2 w 1 0]
[0 1 0 w^2 w 1 w^2 1 0 1 w^2 1 w w^2 0]
[0 0 1 w 1 1 1 1 w 1 0 0 w^2 w 0]

[16, 3, 11] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w 1 w 0 w^2 w w w w 1]
[0 1 0 1 1 w^2 1 w 1 w^2 0 0 1 w^2 w^2 w^2]
[0 0 1 1 w 1 1 w w^2 w w^2 1 w w 0 0]

[17, 3, 12] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 w^2 w^2 w^2 w^2 w^2 w w^2 1 1 1 0 w]
[0 1 0 w^2 w^2 w^2 w^2 1 w 0 w^2 0 0 w w w 1]
[0 0 1 1 w w 1 w 0 w^2 w w w 0 w 1 1]

[18, 3, 13] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w 1 w 0 w^2 w w w 0 w 1 1 0]
[0 1 0 1 1 w^2 w 1 w^2 0 0 1 w^2 1 w^2 1 w^2 w^2]
[0 0 1 1 w 1 w w^2 w w^2 1 w w w 0 w^2 0 w^2]

[19, 3, 14] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w 1 w 0 w^2 w w w 0 w 1 1 0]
[0 1 0 1 1 w^2 1 w 1 w^2 0 0 1 w^2 1 w^2 1 w^2 w^2]
[0 0 1 1 w 1 1 w w^2 w w^2 1 w w w 0 w^2 0 w^2]

[20, 3, 14] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w 1 w 0 w^2 w w w 0 w 1 1 0 0]
[0 1 0 1 1 w^2 1 w 1 w^2 0 0 1 w^2 1 w^2 1 w^2 w^2 0]
[0 0 1 1 w 1 1 w w^2 w w^2 1 w w w 0 w^2 0 w^2 0]

[21, 3, 15] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w 1 w 0 w^2 w w w 0 w 1 1 0 0 1]
[0 1 0 1 1 w^2 1 w 1 w^2 0 0 1 w^2 1 w^2 1 w^2 w^2 0 0]
[0 0 1 1 w 1 1 w w^2 w w^2 1 w w w 0 w^2 0 w^2 1 w]

[22, 3, 16] Linear Code over GF(2^2)
Generator matrix:
[1 0 w^2 w^2 w w 0 1 w w^2 1 0 w^2 w w w^2 0 1 w^2 w 1 0]
[0 1 w w w^2 w^2 0 1 w w^2 0 1 w w^2 0 1 w w^2 0 1 w w^2]
[0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[23, 3, 16] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 w w w w w^2 w^2 w^2 w^2]
[0 0 1 0 1 w w^2 0 1 w w^2 0 1 w w^2 0 1 w w^2 0 1 w w^2]

[24, 3, 17] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w^2 w^2 w w w w w^2 w^2 w^2 0 0 1 w w w 1 1 0 0 0]
[0 1 0 w 0 1 1 w w 1 0 w^2 w 1 w^2 1 w 0 1 w 0 1 w^2 w]
[0 0 1 0 w w^2 w 1 w 1 1 w w^2 w^2 1 w w^2 1 0 0 w w 0 1]

[25, 3, 18] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w^2 w^2 w w w^2 w^2 w w w 1 1 0 1 1 1 w w w^2 w^2 w^2 w]
[0 1 0 w 0 1 1 w w 1 0 w^2 w 1 w^2 1 w 0 1 w 0 1 w^2 w 0]
[0 0 1 0 w w^2 w 1 w^2 0 0 w^2 w w 0 w^2 0 w w^2 w^2 1 1 w^2 w w]

[26, 3, 18] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w^2 w^2 w w w^2 w^2 w w w 1 1 0 1 1 1 w w w^2 w^2 w^2 w 0]
[0 1 0 w 0 1 1 w w 1 0 w^2 w 1 w^2 1 w 0 1 w 0 1 w^2 w 0 0]
[0 0 1 0 w w^2 w 1 w^2 0 0 w^2 w w 0 w^2 0 w w^2 w^2 1 1 w^2 w w 0]

[27, 3, 19] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 w 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[28, 3, 20] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w w 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[29, 3, 21] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 w w^2 w w^2 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w^2 w 1 0 w^2 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 w^2 1 1 w 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[30, 3, 22] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w^2 1 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 1 w^2 w^2 w 0 w 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 w w 0 w^2 w^2 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[31, 3, 22] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w^2 1 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0 0]
[0 1 0 1 w^2 w^2 w 0 w 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1 0]
[0 0 1 w w 0 w^2 w^2 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0]

[32, 3, 23] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 1 w 1 w w 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 1 0 1 1 w w w^2 w^2 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[33, 3, 24] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 w w^2 w^2 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w^2 w 1 w^2 1 1 w^2 1 w^2 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 w 1 1 1 1 1 0 0 w 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[34, 3, 25] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 w^2 w 1 w w^2 0 w w^2 w^2 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 1 1 0 0 1 1 w w w^2 w^2 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[35, 3, 26] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 1 w w^2 w^2 w 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w^2 w 1 w^2 1 0 1 w^2 1 w w^2 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 w 1 1 1 1 w 1 0 0 w^2 w 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[36, 3, 26] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 0 1 1 w 0 1 w w^2 w^2 w 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0 0]
[0 1 0 w^2 w 1 w^2 1 0 1 w^2 1 w w^2 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1 0]
[0 0 1 w 1 1 1 1 w 1 0 0 w^2 w 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0]

[48, 3, 35] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 w 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 w 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

[52, 3, 38] Linear Code over GF(2^2)
Generator matrix:
[1 0 0 1 1 w w^2 1 1 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0 0 1 0 w^2 w 1 0 w^2 w 1 w^2 0 1 w w 1 0 w^2 1 w w^2 0]
[0 1 0 1 w^2 w^2 w 0 w 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1 0 0 1 1 1 1 0 0 0 0 w^2 w^2 w^2 w^2 w w w w 1 1 1 1]
[0 0 1 w w 0 w^2 w^2 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]