Best Known (59, 77, s)-Nets in Base 7
(59, 77, 1868)-Net over F7 — Constructive and digital
Digital (59, 77, 1868)-net over F7, using
- 71 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(776, 1868, F7, 18, 18) (dual of [(1868, 18), 33548, 19]-NRT-code), using
(59, 77, 11702)-Net over F7 — Digital
Digital (59, 77, 11702)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(777, 11702, F7, 18) (dual of [11702, 11625, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(777, 16818, F7, 18) (dual of [16818, 16741, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(777, 16818, F7, 18) (dual of [16818, 16741, 19]-code), using
(59, 77, large)-Net in Base 7 — Upper bound on s
There is no (59, 77, large)-net in base 7, because
- 16 times m-reduction [i] would yield (59, 61, large)-net in base 7, but