Best Known (70, 70+13, s)-Nets in Base 25
(70, 70+13, 1528311)-Net over F25 — Constructive and digital
Digital (70, 83, 1528311)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (16, 22, 130211)-net over F25, using
- net defined by OOA [i] based on linear OOA(2522, 130211, F25, 6, 6) (dual of [(130211, 6), 781244, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2522, 390633, F25, 6) (dual of [390633, 390611, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2522, 390634, F25, 6) (dual of [390634, 390612, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(2521, 390625, F25, 6) (dual of [390625, 390604, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2522, 390634, F25, 6) (dual of [390634, 390612, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2522, 390633, F25, 6) (dual of [390633, 390611, 7]-code), using
- net defined by OOA [i] based on linear OOA(2522, 130211, F25, 6, 6) (dual of [(130211, 6), 781244, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (16, 22, 130211)-net over F25, using
(70, 70+13, large)-Net over F25 — Digital
Digital (70, 83, large)-net over F25, using
- t-expansion [i] based on digital (69, 83, large)-net over F25, using
- 4 times m-reduction [i] based on digital (69, 87, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
- 4 times m-reduction [i] based on digital (69, 87, large)-net over F25, using
(70, 70+13, large)-Net in Base 25 — Upper bound on s
There is no (70, 83, large)-net in base 25, because
- 11 times m-reduction [i] would yield (70, 72, large)-net in base 25, but