Best Known (129−17, 129, s)-Nets in Base 5
(129−17, 129, 244157)-Net over F5 — Constructive and digital
Digital (112, 129, 244157)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (101, 118, 244141)-net over F5, using
- net defined by OOA [i] based on linear OOA(5118, 244141, F5, 17, 17) (dual of [(244141, 17), 4150279, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5118, 1953129, F5, 17) (dual of [1953129, 1953011, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5118, 1953134, F5, 17) (dual of [1953134, 1953016, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5118, 1953134, F5, 17) (dual of [1953134, 1953016, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5118, 1953129, F5, 17) (dual of [1953129, 1953011, 18]-code), using
- net defined by OOA [i] based on linear OOA(5118, 244141, F5, 17, 17) (dual of [(244141, 17), 4150279, 18]-NRT-code), using
- digital (3, 11, 16)-net over F5, using
(129−17, 129, 1479952)-Net over F5 — Digital
Digital (112, 129, 1479952)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5129, 1479952, F5, 17) (dual of [1479952, 1479823, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 1953137, F5, 17) (dual of [1953137, 1953008, 18]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(55, 6, F5, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,5)), using
- dual of repetition code with length 6 [i]
- linear OA(56, 6, F5, 6) (dual of [6, 0, 7]-code or 6-arc in PG(5,5)), using
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(55, 6, F5, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,5)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5129, 1953137, F5, 17) (dual of [1953137, 1953008, 18]-code), using
(129−17, 129, large)-Net in Base 5 — Upper bound on s
There is no (112, 129, large)-net in base 5, because
- 15 times m-reduction [i] would yield (112, 114, large)-net in base 5, but