Best Known (111−24, 111, s)-Nets in Base 4
(111−24, 111, 1042)-Net over F4 — Constructive and digital
Digital (87, 111, 1042)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (72, 96, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (3, 15, 14)-net over F4, using
(111−24, 111, 3073)-Net over F4 — Digital
Digital (87, 111, 3073)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 3073, F4, 24) (dual of [3073, 2962, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 4111, F4, 24) (dual of [4111, 4000, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(41, 14, F4, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4111, 4111, F4, 24) (dual of [4111, 4000, 25]-code), using
(111−24, 111, 653564)-Net in Base 4 — Upper bound on s
There is no (87, 111, 653565)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 739990 495125 473159 859115 983961 372921 628720 551033 965442 566838 354423 > 4111 [i]