Best Known (126, 126+20, s)-Nets in Base 5
(126, 126+20, 195314)-Net over F5 — Constructive and digital
Digital (126, 146, 195314)-net over F5, using
- net defined by OOA [i] based on linear OOA(5146, 195314, F5, 20, 20) (dual of [(195314, 20), 3906134, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(5146, 1953140, F5, 20) (dual of [1953140, 1952994, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 1953144, F5, 20) (dual of [1953144, 1952998, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(51, 19, F5, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 1953144, F5, 20) (dual of [1953144, 1952998, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(5146, 1953140, F5, 20) (dual of [1953140, 1952994, 21]-code), using
(126, 126+20, 976572)-Net over F5 — Digital
Digital (126, 146, 976572)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5146, 976572, F5, 2, 20) (dual of [(976572, 2), 1952998, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5146, 1953144, F5, 20) (dual of [1953144, 1952998, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(51, 19, F5, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(5146, 1953144, F5, 20) (dual of [1953144, 1952998, 21]-code), using
(126, 126+20, large)-Net in Base 5 — Upper bound on s
There is no (126, 146, large)-net in base 5, because
- 18 times m-reduction [i] would yield (126, 128, large)-net in base 5, but