Best Known (227−40, 227, s)-Nets in Base 4
(227−40, 227, 1553)-Net over F4 — Constructive and digital
Digital (187, 227, 1553)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 23, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (164, 204, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 68, 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, 68, 513)-net over F64, using
- digital (3, 23, 14)-net over F4, using
(227−40, 227, 16446)-Net over F4 — Digital
Digital (187, 227, 16446)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4227, 16446, F4, 40) (dual of [16446, 16219, 41]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4224, 16440, F4, 40) (dual of [16440, 16216, 41]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(4211, 16385, F4, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- 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(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(4224, 16443, F4, 39) (dual of [16443, 16219, 40]-code), using Gilbert–Varšamov bound and bm = 4224 > Vbs−1(k−1) = 39 821365 238914 168531 951313 719462 111168 843573 725219 539111 694287 426719 268487 085353 314936 884335 370330 497102 072402 578233 559990 809587 789556 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4224, 16440, F4, 40) (dual of [16440, 16216, 41]-code), using
- construction X with Varšamov bound [i] based on
(227−40, 227, large)-Net in Base 4 — Upper bound on s
There is no (187, 227, large)-net in base 4, because
- 38 times m-reduction [i] would yield (187, 189, large)-net in base 4, but