Best Known (154, 187, s)-Nets in Base 4
(154, 187, 1539)-Net over F4 — Constructive and digital
Digital (154, 187, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (154, 189, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 63, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 63, 513)-net over F64, using
(154, 187, 16448)-Net over F4 — Digital
Digital (154, 187, 16448)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4187, 16448, F4, 33) (dual of [16448, 16261, 34]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4183, 16441, F4, 33) (dual of [16441, 16258, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4183, 16444, F4, 31) (dual of [16444, 16261, 32]-code), using Gilbert–Varšamov bound and bm = 4183 > Vbs−1(k−1) = 2 281275 752639 839628 473199 693768 989594 550212 751797 287055 435974 830411 787802 510662 529396 886697 416272 676866 116904 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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
- linear OA(4183, 16441, F4, 33) (dual of [16441, 16258, 34]-code), using
- construction X with Varšamov bound [i] based on
(154, 187, large)-Net in Base 4 — Upper bound on s
There is no (154, 187, large)-net in base 4, because
- 31 times m-reduction [i] would yield (154, 156, large)-net in base 4, but