Best Known (76, 76+14, s)-Nets in Base 7
(76, 76+14, 117653)-Net over F7 — Constructive and digital
Digital (76, 90, 117653)-net over F7, using
- 1 times m-reduction [i] based on digital (76, 91, 117653)-net over F7, using
- net defined by OOA [i] based on linear OOA(791, 117653, F7, 15, 15) (dual of [(117653, 15), 1764704, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(791, 823572, F7, 15) (dual of [823572, 823481, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 823577, F7, 15) (dual of [823577, 823486, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(76, 34, F7, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(791, 823577, F7, 15) (dual of [823577, 823486, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(791, 823572, F7, 15) (dual of [823572, 823481, 16]-code), using
- net defined by OOA [i] based on linear OOA(791, 117653, F7, 15, 15) (dual of [(117653, 15), 1764704, 16]-NRT-code), using
(76, 76+14, 823577)-Net over F7 — Digital
Digital (76, 90, 823577)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(790, 823577, F7, 14) (dual of [823577, 823487, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(789, 823575, F7, 14) (dual of [823575, 823486, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(789, 823576, F7, 13) (dual of [823576, 823487, 14]-code), using Gilbert–Varšamov bound and bm = 789 > Vbs−1(k−1) = 442468 288632 161639 166543 598575 464477 845611 885138 518837 347802 399263 838071 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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
- linear OA(789, 823575, F7, 14) (dual of [823575, 823486, 15]-code), using
- construction X with Varšamov bound [i] based on
(76, 76+14, large)-Net in Base 7 — Upper bound on s
There is no (76, 90, large)-net in base 7, because
- 12 times m-reduction [i] would yield (76, 78, large)-net in base 7, but