Best Known (77, 98, s)-Nets in Base 4
(77, 98, 1043)-Net over F4 — Constructive and digital
Digital (77, 98, 1043)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-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 (4, 14, 15)-net over F4, using
(77, 98, 3117)-Net over F4 — Digital
Digital (77, 98, 3117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(498, 3117, F4, 21) (dual of [3117, 3019, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(498, 4110, F4, 21) (dual of [4110, 4012, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(497, 4097, F4, 21) (dual of [4097, 4000, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(485, 4097, F4, 19) (dual of [4097, 4012, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(498, 4110, F4, 21) (dual of [4110, 4012, 22]-code), using
(77, 98, 1044320)-Net in Base 4 — Upper bound on s
There is no (77, 98, 1044321)-net in base 4, because
- 1 times m-reduction [i] would yield (77, 97, 1044321)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 25108 582892 688148 455027 751967 703980 496199 439950 736394 272656 > 497 [i]