Best Known (136−19, 136, s)-Nets in Base 5
(136−19, 136, 217014)-Net over F5 — Constructive and digital
Digital (117, 136, 217014)-net over F5, using
- net defined by OOA [i] based on linear OOA(5136, 217014, F5, 19, 19) (dual of [(217014, 19), 4123130, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5136, 1953127, F5, 19) (dual of [1953127, 1952991, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5136, 1953127, F5, 19) (dual of [1953127, 1952991, 20]-code), using
(136−19, 136, 976567)-Net over F5 — Digital
Digital (117, 136, 976567)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5136, 976567, F5, 2, 19) (dual of [(976567, 2), 1952998, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
(136−19, 136, large)-Net in Base 5 — Upper bound on s
There is no (117, 136, large)-net in base 5, because
- 17 times m-reduction [i] would yield (117, 119, large)-net in base 5, but