Best Known (28, 37, s)-Nets in Base 7
(28, 37, 4203)-Net over F7 — Constructive and digital
Digital (28, 37, 4203)-net over F7, using
- net defined by OOA [i] based on linear OOA(737, 4203, F7, 9, 9) (dual of [(4203, 9), 37790, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(737, 16813, F7, 9) (dual of [16813, 16776, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(736, 16812, F7, 9) (dual of [16812, 16776, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(736, 16812, F7, 9) (dual of [16812, 16776, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(737, 16813, F7, 9) (dual of [16813, 16776, 10]-code), using
(28, 37, 12499)-Net over F7 — Digital
Digital (28, 37, 12499)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(737, 12499, F7, 9) (dual of [12499, 12462, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(737, 16813, F7, 9) (dual of [16813, 16776, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(736, 16812, F7, 9) (dual of [16812, 16776, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(736, 16812, F7, 9) (dual of [16812, 16776, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(737, 16813, F7, 9) (dual of [16813, 16776, 10]-code), using
(28, 37, large)-Net in Base 7 — Upper bound on s
There is no (28, 37, large)-net in base 7, because
- 7 times m-reduction [i] would yield (28, 30, large)-net in base 7, but