Best Known (31, 42, s)-Nets in Base 4
(31, 42, 514)-Net over F4 — Constructive and digital
Digital (31, 42, 514)-net over F4, using
- trace code for nets [i] based on digital (10, 21, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(10,256) in PG(20,16)) for nets [i] based on digital (0, 11, 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
- base reduction for projective spaces (embedding PG(10,256) in PG(20,16)) for nets [i] based on digital (0, 11, 257)-net over F256, using
(31, 42, 758)-Net over F4 — Digital
Digital (31, 42, 758)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(442, 758, F4, 11) (dual of [758, 716, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(442, 1036, F4, 11) (dual of [1036, 994, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(441, 1025, F4, 11) (dual of [1025, 984, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(442, 1036, F4, 11) (dual of [1036, 994, 12]-code), using
(31, 42, 75090)-Net in Base 4 — Upper bound on s
There is no (31, 42, 75091)-net in base 4, because
- 1 times m-reduction [i] would yield (31, 41, 75091)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4 835824 094478 363618 591448 > 441 [i]