Best Known (59−16, 59, s)-Nets in Base 7
(59−16, 59, 302)-Net over F7 — Constructive and digital
Digital (43, 59, 302)-net over F7, using
- 72 times duplication [i] based on digital (41, 57, 302)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 302, F7, 16, 16) (dual of [(302, 16), 4775, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(757, 2416, F7, 16) (dual of [2416, 2359, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 16, F7, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(757, 2417, F7, 16) (dual of [2417, 2360, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(757, 2416, F7, 16) (dual of [2416, 2359, 17]-code), using
- net defined by OOA [i] based on linear OOA(757, 302, F7, 16, 16) (dual of [(302, 16), 4775, 17]-NRT-code), using
(59−16, 59, 2537)-Net over F7 — Digital
Digital (43, 59, 2537)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(759, 2537, F7, 16) (dual of [2537, 2478, 17]-code), using
- 126 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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(15) ⊂ Ce(14) [i] based on
- 126 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
(59−16, 59, 1071852)-Net in Base 7 — Upper bound on s
There is no (43, 59, 1071853)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 72 574987 420651 189423 730009 159593 206811 935816 314881 > 759 [i]