Best Known (193−35, 193, s)-Nets in Base 4
(193−35, 193, 1539)-Net over F4 — Constructive and digital
Digital (158, 193, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (158, 195, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 65, 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, 65, 513)-net over F64, using
(193−35, 193, 13941)-Net over F4 — Digital
Digital (158, 193, 13941)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4193, 13941, F4, 35) (dual of [13941, 13748, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 16423, F4, 35) (dual of [16423, 16230, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- linear OA(4183, 16385, F4, 35) (dual of [16385, 16202, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4155, 16385, F4, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(410, 38, F4, 5) (dual of [38, 28, 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 C([0,17]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4193, 16423, F4, 35) (dual of [16423, 16230, 36]-code), using
(193−35, 193, large)-Net in Base 4 — Upper bound on s
There is no (158, 193, large)-net in base 4, because
- 33 times m-reduction [i] would yield (158, 160, large)-net in base 4, but