Best Known (111, 111+17, s)-Nets in Base 5
(111, 111+17, 244153)-Net over F5 — Constructive and digital
Digital (111, 128, 244153)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 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 (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 (2, 10, 12)-net over F5, using
(111, 111+17, 1329381)-Net over F5 — Digital
Digital (111, 128, 1329381)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5128, 1329381, F5, 17) (dual of [1329381, 1329253, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 1953138, F5, 17) (dual of [1953138, 1953010, 18]-code), using
- (u, u+v)-construction [i] based on
- linear OA(510, 13, F5, 8) (dual of [13, 3, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(510, 15, F5, 8) (dual of [15, 5, 9]-code), 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(510, 13, F5, 8) (dual of [13, 3, 9]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5128, 1953138, F5, 17) (dual of [1953138, 1953010, 18]-code), using
(111, 111+17, large)-Net in Base 5 — Upper bound on s
There is no (111, 128, large)-net in base 5, because
- 15 times m-reduction [i] would yield (111, 113, large)-net in base 5, but