Best Known (165, 165+36, s)-Nets in Base 4
(165, 165+36, 1539)-Net over F4 — Constructive and digital
Digital (165, 201, 1539)-net over F4, using
- t-expansion [i] based on digital (164, 201, 1539)-net over F4, using
- 3 times m-reduction [i] based on 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
- 3 times m-reduction [i] based on digital (164, 204, 1539)-net over F4, using
(165, 165+36, 15672)-Net over F4 — Digital
Digital (165, 201, 15672)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4201, 15672, F4, 36) (dual of [15672, 15471, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 16430, F4, 36) (dual of [16430, 16229, 37]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4200, 16429, F4, 36) (dual of [16429, 16229, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(29) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(410, 45, F4, 5) (dual of [45, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(36) ⊂ Ce(29) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4200, 16429, F4, 36) (dual of [16429, 16229, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 16430, F4, 36) (dual of [16430, 16229, 37]-code), using
(165, 165+36, large)-Net in Base 4 — Upper bound on s
There is no (165, 201, large)-net in base 4, because
- 34 times m-reduction [i] would yield (165, 167, large)-net in base 4, but