Best Known (83, 83+13, s)-Nets in Base 7
(83, 83+13, 960814)-Net over F7 — Constructive and digital
Digital (83, 96, 960814)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (76, 89, 960801)-net over F7, using
- net defined by OOA [i] based on linear OOA(789, 960801, F7, 13, 13) (dual of [(960801, 13), 12490324, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(789, 5764807, F7, 13) (dual of [5764807, 5764718, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(789, 5764809, F7, 13) (dual of [5764809, 5764720, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(789, 5764809, F7, 13) (dual of [5764809, 5764720, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(789, 5764807, F7, 13) (dual of [5764807, 5764718, 14]-code), using
- net defined by OOA [i] based on linear OOA(789, 960801, F7, 13, 13) (dual of [(960801, 13), 12490324, 14]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(83, 83+13, 5764848)-Net over F7 — Digital
Digital (83, 96, 5764848)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(796, 5764848, F7, 13) (dual of [5764848, 5764752, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(789, 5764801, F7, 13) (dual of [5764801, 5764712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(77, 47, F7, 4) (dual of [47, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 48, F7, 4) (dual of [48, 41, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(77, 48, F7, 4) (dual of [48, 41, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
(83, 83+13, large)-Net in Base 7 — Upper bound on s
There is no (83, 96, large)-net in base 7, because
- 11 times m-reduction [i] would yield (83, 85, large)-net in base 7, but