Best Known (24, 24+9, s)-Nets in Base 7
(24, 24+9, 609)-Net over F7 — Constructive and digital
Digital (24, 33, 609)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (20, 29, 601)-net over F7, using
- net defined by OOA [i] based on linear OOA(729, 601, F7, 9, 9) (dual of [(601, 9), 5380, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(729, 2405, F7, 9) (dual of [2405, 2376, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(729, 2401, F7, 9) (dual of [2401, 2372, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(729, 2405, F7, 9) (dual of [2405, 2376, 10]-code), using
- net defined by OOA [i] based on linear OOA(729, 601, F7, 9, 9) (dual of [(601, 9), 5380, 10]-NRT-code), using
- digital (0, 4, 8)-net over F7, using
(24, 24+9, 2547)-Net over F7 — Digital
Digital (24, 33, 2547)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(733, 2547, F7, 9) (dual of [2547, 2514, 10]-code), using
- 138 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 30 times 0, 1, 100 times 0) [i] based on linear OA(729, 2405, F7, 9) (dual of [2405, 2376, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(729, 2401, F7, 9) (dual of [2401, 2372, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 138 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 30 times 0, 1, 100 times 0) [i] based on linear OA(729, 2405, F7, 9) (dual of [2405, 2376, 10]-code), using
(24, 24+9, 2126598)-Net in Base 7 — Upper bound on s
There is no (24, 33, 2126599)-net in base 7, because
- 1 times m-reduction [i] would yield (24, 32, 2126599)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1104 429379 497813 276965 702377 > 732 [i]