Best Known (21, 21+9, s)-Nets in Base 7
(21, 21+9, 601)-Net over F7 — Constructive and digital
Digital (21, 30, 601)-net over F7, using
- 71 times duplication [i] based on 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
(21, 21+9, 1782)-Net over F7 — Digital
Digital (21, 30, 1782)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(730, 1782, F7, 9) (dual of [1782, 1752, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(730, 2406, F7, 9) (dual of [2406, 2376, 10]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(729, 2405, F7, 9) (dual of [2405, 2376, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(730, 2406, F7, 9) (dual of [2406, 2376, 10]-code), using
(21, 21+9, 494151)-Net in Base 7 — Upper bound on s
There is no (21, 30, 494152)-net in base 7, because
- 1 times m-reduction [i] would yield (21, 29, 494152)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 3 219909 062517 430070 615233 > 729 [i]