Best Known (109, 109+20, s)-Nets in Base 5
(109, 109+20, 39063)-Net over F5 — Constructive and digital
Digital (109, 129, 39063)-net over F5, using
- net defined by OOA [i] based on linear OOA(5129, 39063, F5, 20, 20) (dual of [(39063, 20), 781131, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(5129, 390630, F5, 20) (dual of [390630, 390501, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 390633, F5, 20) (dual of [390633, 390504, 21]-code), using
- 1 times truncation [i] based on linear OA(5130, 390634, F5, 21) (dual of [390634, 390504, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(18) [i] based on
- 1 times truncation [i] based on linear OA(5130, 390634, F5, 21) (dual of [390634, 390504, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 390633, F5, 20) (dual of [390633, 390504, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(5129, 390630, F5, 20) (dual of [390630, 390501, 21]-code), using
(109, 109+20, 195316)-Net over F5 — Digital
Digital (109, 129, 195316)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5129, 195316, F5, 2, 20) (dual of [(195316, 2), 390503, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5129, 390632, F5, 20) (dual of [390632, 390503, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 390633, F5, 20) (dual of [390633, 390504, 21]-code), using
- 1 times truncation [i] based on linear OA(5130, 390634, F5, 21) (dual of [390634, 390504, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(18) [i] based on
- 1 times truncation [i] based on linear OA(5130, 390634, F5, 21) (dual of [390634, 390504, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5129, 390633, F5, 20) (dual of [390633, 390504, 21]-code), using
- OOA 2-folding [i] based on linear OA(5129, 390632, F5, 20) (dual of [390632, 390503, 21]-code), using
(109, 109+20, large)-Net in Base 5 — Upper bound on s
There is no (109, 129, large)-net in base 5, because
- 18 times m-reduction [i] would yield (109, 111, large)-net in base 5, but