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Table 1 New Quantum codes over \(S_{k}\)

From: Quantum codes from constacyclic codes over \(S_{k}\)

n

k

\((\lambda _{1},\ldots , \lambda _{2k+1})\)

\(\langle g_{1}(x),\ldots ,g_{2k+1}(x)\rangle \)

\(\varphi _{k}(C)\)

\([[n,l,d]]_{q}\)

\([[n',l',d']]_{q}\)

8

1

(1,1,−1)

(112,112,1022)

[24,16,3]

\([[24,8, \geq 3]]_{3}\)

\([[24,8, 2]]_{3}\) [13]

24

1

(1,1,1)

(1101,11,11)

[72,67,3]

\([[72,62, \geq 3]]_{3}\)

\([[72,48, 2]]_{3}\) [13]

26

1

(1,1,1)

(101102,121,121)

[78,66,4]

\([[78,54,\geq 4]]_{3}\)

\([[78,48,4]]_{3}\) [17]

12

1

(1,1,1)

(1111,11,11)

[36,31,4]

\([[36,26,\geq 4]]_{3}\)

\([[36,24,3]]_{3}\) [17]

28

1

(1,1,1)

(1111,11,11)

[84,79,4]

\([[84,75,\geq 4]]_{7}\)

\([[84,72,3]]_{7}\) [17]

16

1

(1,1,1)

\((1\omega ^{2}\omega ^{3}\omega ^{5},1\omega ^{2},1\omega ^{2})\)

[48,43,3]

\([[48,38,\geq 3]]_{9}\)

\([[48,30,3]]_{9}\) [16]