Best Known (93−16, 93, s)-Nets in Base 4
(93−16, 93, 2057)-Net over F4 — Constructive and digital
Digital (77, 93, 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, 84, 2048)-net over F4, using
- net defined by OOA [i] based on linear OOA(484, 2048, F4, 16, 16) (dual of [(2048, 16), 32684, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(484, 16384, F4, 16) (dual of [16384, 16300, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(484, 16384, F4, 16) (dual of [16384, 16300, 17]-code), using
- net defined by OOA [i] based on linear OOA(484, 2048, F4, 16, 16) (dual of [(2048, 16), 32684, 17]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(93−16, 93, 16422)-Net over F4 — Digital
Digital (77, 93, 16422)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(493, 16422, F4, 16) (dual of [16422, 16329, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(492, 16420, F4, 16) (dual of [16420, 16328, 17]-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(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(492, 16421, F4, 15) (dual of [16421, 16329, 16]-code), using Gilbert–Varšamov bound and bm = 492 > Vbs−1(k−1) = 5 652354 159452 762794 943295 205529 835421 377288 577081 100392 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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(492, 16420, F4, 16) (dual of [16420, 16328, 17]-code), using
- construction X with Varšamov bound [i] based on
(93−16, 93, large)-Net in Base 4 — Upper bound on s
There is no (77, 93, large)-net in base 4, because
- 14 times m-reduction [i] would yield (77, 79, large)-net in base 4, but