All unmarked upper bounds in the table come from the linear
programming bound of Theorem 21. (A few of these bounds can also
be obtained from Eq. (14) or from Theorem 16.) Unmarked
lower bounds are from Theorem 6.
| a |
The hexacode, a (6,26,4) classical code that can be taken to be the GF(4) span of
<001111, 0101ww2, 1001w2w>
(see Chapter 3 of [CoSl93]).
Aut(h6)=3.S6, of order 2160.
|
| b |
A classical self-dual code over GF(4) -
see [MOSW78],
[CPS79].
|
| c |
A cyclic code, see Table I;
Cylic additive codes [[14,2,5]], [[21,7,5]]
(see here)
|
| d |
A [[25,1,9]] code obtained by concatenating the [[5,1,3]] Hamming code with itself (Fig. 1 of Section 4).
|
| e |
The dodecacode defined in Section 6.
|
| f |
An [[8,3,3]] code, discovered independently in
[CRSS97],
[Got96],
and [Ste96].
The (8,25) additive code may be generated by vectors ((01www21w2))0,
11111111, wwwwwwww (where the double parentheses mean that all
cyclic shifts of the enclosed string are to be used).
Exhaustive search shows that this code is unique.
Another version is obtained from Theorem 10.
The automorphism group has order 168, and is the semidirect product of a cyclic group of order 3 and the general
affine group AGL(1,8)={ x \to ax+b : a,b,x \in GF(8), a \neq 0 }.
|
| g |
A quasicyclic code found by T. A. Gulliver - see Table II of Section 5.
|
| h |
A Hamming code, see Section 5.
|
| i |
Use the (12,28) and (14,28) linear codes with generator matrices
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | w | w |
| 0 | 1 | 0 | 1 | w | w2 | 0 | 1 | 0 | w | 1 | w |
| 1 | 0 | 0 | 1 | w2 | w | 0 | 1 | w | 0 | w | 1 |
| and |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | w | w2 | 0 | 1 | w | w2 | 0 | 1 | w | w2 |
| 1 | 0 | 0 | 1 | w2 | w | 0 | 1 | w2 | w | 1 | 0 | w | w2 |
|
respectively.
Their automorphism groups have orders 720 and 8064, and both act
transitively on the coordinates.
The first of these can be obtained from the u|u+v construction
(c.f. Theorem 12) applied to the unique [[6,4,2]] and [[6,0,4]]
codes.
|
| j |
A [[17,9,4]] code, for which the corresponding (17,28,12) code C is a well-known
linear code, a two-weight code of class TF3 [CaKa86].
The columns of the generator matrix of C represent the 17 points of an ovoid in PG(3,4).
Both C and C^\perp are cyclic, a generator for C^\perp being 1w1w1000000000000.
The weight distribution of C is A0=1,
A12=204, A16=51, and its automorphism group has order 48 960.
|
| k |
The [[16,4,5]] extended cyclic code spanned by
((w^2w^20w1w1111001111))0, together with vectors of
all 1's and all w's .
|
| n |
A [[32,12,6]] code obtained from an extended BCH code over
GF(16).
|
| r |
From classical linear codes [40,7,24] and [45,10,18] via Rains' puncture code.
|
| s |
By shortening one of the following codes using Theorem 7 or its additive analogue:
the [[21,15,3]] or [[85,77,3]] Hamming codes (see Section 5),
the [[32,25,3]] Gottesman code (Theorem 10),
the [[40,30,4]] code given in Table II or
[[40,33,3]] code shown in Fig. 2.
Additionally, there is a linear quantum code [[41,31,4]]
found by Vladimir Tonchev that can be shortened to codes of length
n=10, 12, 14, 15, 16, 17, 18, 19, 20,21, 22, 23, 24, 25, 26, 27,
29, 31, 33, 35, 41.
|
| t |
A quantum twisted code (see J. Bierbrauer and Y. Edel)
|
| u |
From the u|u+v construction (see Theorem 12).
|
| v |
The following (17,26) code with trivial automorphism group found by random search:
| 0 | 0 | 1 | 0 | w | w2 | w | w2 | 1 | 1 | w | w2 | 0 | 0 | 1 | 1 | w2 |
| 0 | 0 | w | 1 | 0 | w | 0 | w2 | w2 | w2 | 1 | 1 | w | w2 | w2 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 | w | 1 | w | w2 | w2 | w2 | 0 | w2 | 1 | w | 0 | w2 |
| 0 | w | 0 | w | w | 0 | w2 | 1 | w2 | 1 | w | w2 | w | 1 | w | w | 1 |
| 1 | 0 | 0 | w | w2 | 0 | 0 | 1 | w | w | w2 | 1 | w2 | w | 0 | w2 | 1 |
| w | 0 | 0 | 1 | w2 | w2 | w2 | 0 | w2 | 0 | w2 | 1 | 0 | 1 | 1 | w | w2 |
|
| y |
Cyclic codes [[24,3,7]], [[31,0,10]],
[[30,5,8]],[[30,6,7]], [[30,8,7]],
[[30,11,6]], [[31,16,5]], and [[36,3,10]]
|
| z |
Quasi-cyclic codes [[20,4,6]], [[24,8,5]],
[[27,9,6]], [[28,10,6]], [[30,7,16]], [[30,12,6]],
[[32,6,8]], and [[34,24,4]] by M. Grassl and Zlatko Varbanov,
Faculty of Mathematics and Informatics, Veliko Tarnovo University, Bulgaria
|
| li |
A code [[16,7,4]] found by Li Yu, Carnegie-Mellon University,
2007-03-13.
|
| to |
A code [[27,13,15]] found by Vladimir D. Tonchev, Quantum codes from caps, Discrete Mathematics (2008),
doi: 10.1016/j.disc.2007.12.007
|
| mg |
From a cyclic linear code [43,36,5] over GF(4) which does not
contain its Hermitian dual using the puncture code of Eric
Rains, or from a linear code [36,24,8] over GF(4).
|