Best Known (73, 73+14, s)-Nets in Base 7
(73, 73+14, 117651)-Net over F7 — Constructive and digital
Digital (73, 87, 117651)-net over F7, using
- 1 times m-reduction [i] based on digital (73, 88, 117651)-net over F7, using
- net defined by OOA [i] based on linear OOA(788, 117651, F7, 15, 15) (dual of [(117651, 15), 1764677, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(788, 823558, F7, 15) (dual of [823558, 823470, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(788, 823560, F7, 15) (dual of [823560, 823472, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [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(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(73, 17, F7, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(788, 823560, F7, 15) (dual of [823560, 823472, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(788, 823558, F7, 15) (dual of [823558, 823470, 16]-code), using
- net defined by OOA [i] based on linear OOA(788, 117651, F7, 15, 15) (dual of [(117651, 15), 1764677, 16]-NRT-code), using
(73, 73+14, 823561)-Net over F7 — Digital
Digital (73, 87, 823561)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(787, 823561, F7, 14) (dual of [823561, 823474, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(786, 823559, F7, 14) (dual of [823559, 823473, 15]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [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(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(715, 16, F7, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,7)), using
- dual of repetition code with length 16 [i]
- linear OA(71, 16, F7, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(786, 823560, F7, 13) (dual of [823560, 823474, 14]-code), using Gilbert–Varšamov bound and bm = 786 > Vbs−1(k−1) = 442365 146374 795974 290619 180538 801557 303129 087267 146082 667644 870133 709047 [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(786, 823559, F7, 14) (dual of [823559, 823473, 15]-code), using
- construction X with Varšamov bound [i] based on
(73, 73+14, large)-Net in Base 7 — Upper bound on s
There is no (73, 87, large)-net in base 7, because
- 12 times m-reduction [i] would yield (73, 75, large)-net in base 7, but