Best Known (42−13, 42, s)-Nets in Base 5
(42−13, 42, 146)-Net over F5 — Constructive and digital
Digital (29, 42, 146)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- digital (21, 34, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 17, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 17, 66)-net over F25, using
(42−13, 42, 488)-Net over F5 — Digital
Digital (29, 42, 488)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(542, 488, F5, 13) (dual of [488, 446, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(542, 635, F5, 13) (dual of [635, 593, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(541, 626, F5, 13) (dual of [626, 585, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 58−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(533, 626, F5, 11) (dual of [626, 593, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 58−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(542, 635, F5, 13) (dual of [635, 593, 14]-code), using
(42−13, 42, 44711)-Net in Base 5 — Upper bound on s
There is no (29, 42, 44712)-net in base 5, because
- 1 times m-reduction [i] would yield (29, 41, 44712)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 45478 394897 576432 748962 288257 > 541 [i]