Best Known (78, 101, s)-Nets in Base 7
(78, 101, 1529)-Net over F7 — Constructive and digital
Digital (78, 101, 1529)-net over F7, using
- 72 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
(78, 101, 15284)-Net over F7 — Digital
Digital (78, 101, 15284)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7101, 15284, F7, 23) (dual of [15284, 15183, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7101, 16808, F7, 23) (dual of [16808, 16707, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(7101, 16808, F7, 23) (dual of [16808, 16707, 24]-code), using
(78, 101, large)-Net in Base 7 — Upper bound on s
There is no (78, 101, large)-net in base 7, because
- 21 times m-reduction [i] would yield (78, 80, large)-net in base 7, but