Best Known (94−17, 94, s)-Nets in Base 4
(94−17, 94, 2057)-Net over F4 — Constructive and digital
Digital (77, 94, 2057)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (68, 85, 2048)-net over F4, using
- net defined by OOA [i] based on linear OOA(485, 2048, F4, 17, 17) (dual of [(2048, 17), 34731, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using
- net defined by OOA [i] based on linear OOA(485, 2048, F4, 17, 17) (dual of [(2048, 17), 34731, 18]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(94−17, 94, 11561)-Net over F4 — Digital
Digital (77, 94, 11561)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(494, 11561, F4, 17) (dual of [11561, 11467, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(494, 16409, F4, 17) (dual of [16409, 16315, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(457, 16385, F4, 11) (dual of [16385, 16328, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(49, 24, F4, 5) (dual of [24, 15, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 36, F4, 5) (dual of [36, 27, 6]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(494, 16409, F4, 17) (dual of [16409, 16315, 18]-code), using
(94−17, 94, large)-Net in Base 4 — Upper bound on s
There is no (77, 94, large)-net in base 4, because
- 15 times m-reduction [i] would yield (77, 79, large)-net in base 4, but