Best Known (76−19, 76, s)-Nets in Base 4
(76−19, 76, 1028)-Net over F4 — Constructive and digital
Digital (57, 76, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 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
(76−19, 76, 1054)-Net over F4 — Digital
Digital (57, 76, 1054)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(476, 1054, F4, 19) (dual of [1054, 978, 20]-code), using
- 14 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0) [i] based on linear OA(472, 1036, F4, 19) (dual of [1036, 964, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(471, 1025, F4, 19) (dual of [1025, 954, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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([0,9]) ⊂ C([0,8]) [i] based on
- 14 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0) [i] based on linear OA(472, 1036, F4, 19) (dual of [1036, 964, 20]-code), using
(76−19, 76, 143805)-Net in Base 4 — Upper bound on s
There is no (57, 76, 143806)-net in base 4, because
- 1 times m-reduction [i] would yield (57, 75, 143806)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1427 287852 046309 385456 364592 145829 729424 167927 > 475 [i]