Best Known (109−14, 109, s)-Nets in Base 5
(109−14, 109, 279031)-Net over F5 — Constructive and digital
Digital (95, 109, 279031)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (86, 100, 279019)-net over F5, using
- net defined by OOA [i] based on linear OOA(5100, 279019, F5, 14, 14) (dual of [(279019, 14), 3906166, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(5100, 1953133, F5, 14) (dual of [1953133, 1953033, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5100, 1953134, F5, 14) (dual of [1953134, 1953034, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5100, 1953134, F5, 14) (dual of [1953134, 1953034, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(5100, 1953133, F5, 14) (dual of [1953133, 1953033, 15]-code), using
- net defined by OOA [i] based on linear OOA(5100, 279019, F5, 14, 14) (dual of [(279019, 14), 3906166, 15]-NRT-code), using
- digital (2, 9, 12)-net over F5, using
(109−14, 109, 1953179)-Net over F5 — Digital
Digital (95, 109, 1953179)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 1953179, F5, 14) (dual of [1953179, 1953070, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(59, 54, F5, 5) (dual of [54, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
(109−14, 109, large)-Net in Base 5 — Upper bound on s
There is no (95, 109, large)-net in base 5, because
- 12 times m-reduction [i] would yield (95, 97, large)-net in base 5, but