Best Known (91−21, 91, s)-Nets in Base 7
(91−21, 91, 1681)-Net over F7 — Constructive and digital
Digital (70, 91, 1681)-net over F7, using
- net defined by OOA [i] based on linear OOA(791, 1681, F7, 21, 21) (dual of [(1681, 21), 35210, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(791, 16811, F7, 21) (dual of [16811, 16720, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 16812, F7, 21) (dual of [16812, 16721, 22]-code), using
- 1 times truncation [i] based on linear OA(792, 16813, F7, 22) (dual of [16813, 16721, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(71, 6, F7, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(792, 16813, F7, 22) (dual of [16813, 16721, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 16812, F7, 21) (dual of [16812, 16721, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(791, 16811, F7, 21) (dual of [16811, 16720, 22]-code), using
(91−21, 91, 13298)-Net over F7 — Digital
Digital (70, 91, 13298)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(791, 13298, F7, 21) (dual of [13298, 13207, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 16808, F7, 21) (dual of [16808, 16717, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(791, 16808, F7, 21) (dual of [16808, 16717, 22]-code), using
(91−21, 91, large)-Net in Base 7 — Upper bound on s
There is no (70, 91, large)-net in base 7, because
- 19 times m-reduction [i] would yield (70, 72, large)-net in base 7, but