Best Known (58, 76, s)-Nets in Base 7
(58, 76, 1868)-Net over F7 — Constructive and digital
Digital (58, 76, 1868)-net over F7, using
- net defined by OOA [i] based on linear OOA(776, 1868, F7, 18, 18) (dual of [(1868, 18), 33548, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(776, 16812, F7, 18) (dual of [16812, 16736, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(776, 16812, F7, 18) (dual of [16812, 16736, 19]-code), using
(58, 76, 10361)-Net over F7 — Digital
Digital (58, 76, 10361)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(776, 10361, F7, 18) (dual of [10361, 10285, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using
(58, 76, large)-Net in Base 7 — Upper bound on s
There is no (58, 76, large)-net in base 7, because
- 16 times m-reduction [i] would yield (58, 60, large)-net in base 7, but