Best Known (157, 157+33, s)-Nets in Base 4
(157, 157+33, 1539)-Net over F4 — Constructive and digital
Digital (157, 190, 1539)-net over F4, using
- t-expansion [i] based on digital (156, 190, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (156, 192, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 64, 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, 64, 513)-net over F64, using
- 2 times m-reduction [i] based on digital (156, 192, 1539)-net over F4, using
(157, 157+33, 16454)-Net over F4 — Digital
Digital (157, 190, 16454)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4190, 16454, F4, 33) (dual of [16454, 16264, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(22) [i] based on
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(421, 70, F4, 9) (dual of [70, 49, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(419, 64, F4, 9) (dual of [64, 45, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(8) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- construction X applied to Ce(32) ⊂ Ce(22) [i] based on
(157, 157+33, large)-Net in Base 4 — Upper bound on s
There is no (157, 190, large)-net in base 4, because
- 31 times m-reduction [i] would yield (157, 159, large)-net in base 4, but