Best Known (68−9, 68, s)-Nets in Base 5
(68−9, 68, 488289)-Net over F5 — Constructive and digital
Digital (59, 68, 488289)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (55, 64, 488283)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 488283, F5, 9, 9) (dual of [(488283, 9), 4394483, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(564, 1953133, F5, 9) (dual of [1953133, 1953069, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(564, 1953134, F5, 9) (dual of [1953134, 1953070, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(555, 1953125, F5, 8) (dual of [1953125, 1953070, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(564, 1953134, F5, 9) (dual of [1953134, 1953070, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(564, 1953133, F5, 9) (dual of [1953133, 1953069, 10]-code), using
- net defined by OOA [i] based on linear OOA(564, 488283, F5, 9, 9) (dual of [(488283, 9), 4394483, 10]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(68−9, 68, 1953157)-Net over F5 — Digital
Digital (59, 68, 1953157)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(568, 1953157, F5, 9) (dual of [1953157, 1953089, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(567, 1953155, F5, 9) (dual of [1953155, 1953088, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(567, 1953156, F5, 8) (dual of [1953156, 1953089, 9]-code), using Gilbert–Varšamov bound and bm = 567 > Vbs−1(k−1) = 352 486177 137451 133912 994108 051498 624472 670845 [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(567, 1953155, F5, 9) (dual of [1953155, 1953088, 10]-code), using
- construction X with Varšamov bound [i] based on
(68−9, 68, large)-Net in Base 5 — Upper bound on s
There is no (59, 68, large)-net in base 5, because
- 7 times m-reduction [i] would yield (59, 61, large)-net in base 5, but