Best Known (77, 100, s)-Nets in Base 7
(77, 100, 1529)-Net over F7 — Constructive and digital
Digital (77, 100, 1529)-net over F7, using
- 71 times duplication [i] based on digital (76, 99, 1529)-net over F7, using
- net defined by OOA [i] based on linear OOA(799, 1529, F7, 23, 23) (dual of [(1529, 23), 35068, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(799, 16820, F7, 23) (dual of [16820, 16721, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(73, 13, F7, 2) (dual of [13, 10, 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(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(799, 16820, F7, 23) (dual of [16820, 16721, 24]-code), using
- net defined by OOA [i] based on linear OOA(799, 1529, F7, 23, 23) (dual of [(1529, 23), 35068, 24]-NRT-code), using
(77, 100, 13931)-Net over F7 — Digital
Digital (77, 100, 13931)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7100, 13931, F7, 23) (dual of [13931, 13831, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7100, 16826, F7, 23) (dual of [16826, 16726, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(781, 16807, F7, 19) (dual of [16807, 16726, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(74, 19, F7, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,7)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(7100, 16826, F7, 23) (dual of [16826, 16726, 24]-code), using
(77, 100, large)-Net in Base 7 — Upper bound on s
There is no (77, 100, large)-net in base 7, because
- 21 times m-reduction [i] would yield (77, 79, large)-net in base 7, but