Best Known (59, 59+17, s)-Nets in Base 7
(59, 59+17, 2103)-Net over F7 — Constructive and digital
Digital (59, 76, 2103)-net over F7, using
- 71 times duplication [i] based on digital (58, 75, 2103)-net over F7, using
- net defined by OOA [i] based on linear OOA(775, 2103, F7, 17, 17) (dual of [(2103, 17), 35676, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(775, 16825, F7, 17) (dual of [16825, 16750, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(775, 16826, F7, 17) (dual of [16826, 16751, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 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(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(775, 16826, F7, 17) (dual of [16826, 16751, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(775, 16825, F7, 17) (dual of [16825, 16750, 18]-code), using
- net defined by OOA [i] based on linear OOA(775, 2103, F7, 17, 17) (dual of [(2103, 17), 35676, 18]-NRT-code), using
(59, 59+17, 16828)-Net over F7 — Digital
Digital (59, 76, 16828)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(776, 16828, F7, 17) (dual of [16828, 16752, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- 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(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
(59, 59+17, large)-Net in Base 7 — Upper bound on s
There is no (59, 76, large)-net in base 7, because
- 15 times m-reduction [i] would yield (59, 61, large)-net in base 7, but