Best Known (115, 115+18, s)-Nets in Base 5
(115, 115+18, 217017)-Net over F5 — Constructive and digital
Digital (115, 133, 217017)-net over F5, using
- 51 times duplication [i] based on digital (114, 132, 217017)-net over F5, using
- net defined by OOA [i] based on linear OOA(5132, 217017, F5, 18, 18) (dual of [(217017, 18), 3906174, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5132, 1953153, F5, 18) (dual of [1953153, 1953021, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5132, 1953157, F5, 18) (dual of [1953157, 1953025, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5132, 1953157, F5, 18) (dual of [1953157, 1953025, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5132, 1953153, F5, 18) (dual of [1953153, 1953021, 19]-code), using
- net defined by OOA [i] based on linear OOA(5132, 217017, F5, 18, 18) (dual of [(217017, 18), 3906174, 19]-NRT-code), using
(115, 115+18, 993062)-Net over F5 — Digital
Digital (115, 133, 993062)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5133, 993062, F5, 18) (dual of [993062, 992929, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 1953158, F5, 18) (dual of [1953158, 1953025, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5132, 1953157, F5, 18) (dual of [1953157, 1953025, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5132, 1953157, F5, 18) (dual of [1953157, 1953025, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 1953158, F5, 18) (dual of [1953158, 1953025, 19]-code), using
(115, 115+18, large)-Net in Base 5 — Upper bound on s
There is no (115, 133, large)-net in base 5, because
- 16 times m-reduction [i] would yield (115, 117, large)-net in base 5, but