Best Known (99−21, 99, s)-Nets in Base 4
(99−21, 99, 1045)-Net over F4 — Constructive and digital
Digital (78, 99, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (63, 84, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
- digital (5, 15, 17)-net over F4, using
(99−21, 99, 3354)-Net over F4 — Digital
Digital (78, 99, 3354)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 3354, F4, 21) (dual of [3354, 3255, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 4124, F4, 21) (dual of [4124, 4025, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(45, 25, F4, 3) (dual of [25, 20, 4]-code or 25-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 4124, F4, 21) (dual of [4124, 4025, 22]-code), using
(99−21, 99, 1199610)-Net in Base 4 — Upper bound on s
There is no (78, 99, 1199611)-net in base 4, because
- 1 times m-reduction [i] would yield (78, 98, 1199611)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 100434 348661 557532 470815 417745 160371 032886 967963 601725 730803 > 498 [i]