Best Known (97, 97+27, s)-Nets in Base 4
(97, 97+27, 1044)-Net over F4 — Constructive and digital
Digital (97, 124, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 31, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
(97, 97+27, 3090)-Net over F4 — Digital
Digital (97, 124, 3090)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4124, 3090, F4, 27) (dual of [3090, 2966, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4124, 4112, F4, 27) (dual of [4112, 3988, 28]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4122, 4110, F4, 27) (dual of [4110, 3988, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4121, 4097, F4, 27) (dual of [4097, 3976, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4122, 4110, F4, 27) (dual of [4110, 3988, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4124, 4112, F4, 27) (dual of [4112, 3988, 28]-code), using
(97, 97+27, 939061)-Net in Base 4 — Upper bound on s
There is no (97, 124, 939062)-net in base 4, because
- 1 times m-reduction [i] would yield (97, 123, 939062)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 113 079424 244212 126682 572726 043354 952494 996488 483883 900849 961896 930452 643914 > 4123 [i]