Best Known (79−13, 79, s)-Nets in Base 5
(79−13, 79, 13035)-Net over F5 — Constructive and digital
Digital (66, 79, 13035)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- digital (58, 71, 13021)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 13021, F5, 13, 13) (dual of [(13021, 13), 169202, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(571, 78127, F5, 13) (dual of [78127, 78056, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(571, 78132, F5, 13) (dual of [78132, 78061, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(571, 78127, F5, 13) (dual of [78127, 78056, 14]-code), using
- net defined by OOA [i] based on linear OOA(571, 13021, F5, 13, 13) (dual of [(13021, 13), 169202, 14]-NRT-code), using
(79−13, 79, 78162)-Net over F5 — Digital
Digital (66, 79, 78162)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(579, 78162, F5, 13) (dual of [78162, 78083, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(578, 78160, F5, 13) (dual of [78160, 78082, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(578, 78161, F5, 12) (dual of [78161, 78083, 13]-code), using Gilbert–Varšamov bound and bm = 578 > Vbs−1(k−1) = 69830 299833 750382 725278 077292 090410 960130 413663 324865 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(578, 78160, F5, 13) (dual of [78160, 78082, 14]-code), using
- construction X with Varšamov bound [i] based on
(79−13, 79, large)-Net in Base 5 — Upper bound on s
There is no (66, 79, large)-net in base 5, because
- 11 times m-reduction [i] would yield (66, 68, large)-net in base 5, but