Best Known (42−12, 42, s)-Nets in Base 5
(42−12, 42, 208)-Net over F5 — Constructive and digital
Digital (30, 42, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 21, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
(42−12, 42, 676)-Net over F5 — Digital
Digital (30, 42, 676)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(542, 676, F5, 12) (dual of [676, 634, 13]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0, 1, 23 times 0) [i] based on linear OA(537, 629, F5, 12) (dual of [629, 592, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(537, 625, F5, 12) (dual of [625, 588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(533, 625, F5, 11) (dual of [625, 592, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 42 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0, 1, 23 times 0) [i] based on linear OA(537, 629, F5, 12) (dual of [629, 592, 13]-code), using
(42−12, 42, 58468)-Net in Base 5 — Upper bound on s
There is no (30, 42, 58469)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 227383 896345 393946 785473 210249 > 542 [i]