Best Known (38, 38+8, s)-Nets in Base 25
(38, 38+8, 2104964)-Net over F25 — Constructive and digital
Digital (38, 46, 2104964)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (6, 10, 7814)-net over F25, using
- net defined by OOA [i] based on linear OOA(2510, 7814, F25, 4, 4) (dual of [(7814, 4), 31246, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2510, 15628, F25, 4) (dual of [15628, 15618, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(257, 15625, F25, 3) (dual of [15625, 15618, 4]-code or 15625-cap in PG(6,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2510, 15628, F25, 4) (dual of [15628, 15618, 5]-code), using
- net defined by OOA [i] based on linear OOA(2510, 7814, F25, 4, 4) (dual of [(7814, 4), 31246, 5]-NRT-code), using
- digital (28, 36, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- digital (6, 10, 7814)-net over F25, using
(38, 38+8, large)-Net over F25 — Digital
Digital (38, 46, large)-net over F25, using
- t-expansion [i] based on digital (36, 46, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
(38, 38+8, large)-Net in Base 25 — Upper bound on s
There is no (38, 46, large)-net in base 25, because
- 6 times m-reduction [i] would yield (38, 40, large)-net in base 25, but