Best Known (70−21, 70, s)-Nets in Base 5
(70−21, 70, 252)-Net over F5 — Constructive and digital
Digital (49, 70, 252)-net over F5, using
- 8 times m-reduction [i] based on digital (49, 78, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
(70−21, 70, 651)-Net over F5 — Digital
Digital (49, 70, 651)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(570, 651, F5, 21) (dual of [651, 581, 22]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(566, 630, F5, 21) (dual of [630, 564, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(565, 625, F5, 21) (dual of [625, 560, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(561, 625, F5, 19) (dual of [625, 564, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 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 Ce(20) ⊂ Ce(18) [i] based on
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(566, 630, F5, 21) (dual of [630, 564, 22]-code), using
(70−21, 70, 75295)-Net in Base 5 — Upper bound on s
There is no (49, 70, 75296)-net in base 5, because
- 1 times m-reduction [i] would yield (49, 69, 75296)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 694180 390832 307777 997412 930322 258988 437759 132161 > 569 [i]