Best Known (23, 23+9, s)-Nets in Base 25
(23, 23+9, 4220)-Net over F25 — Constructive and digital
Digital (23, 32, 4220)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 314)-net over F25, using
- net defined by OOA [i] based on linear OOA(257, 314, F25, 4, 4) (dual of [(314, 4), 1249, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(257, 628, F25, 4) (dual of [628, 621, 5]-code), using
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(257, 624, F25, 4) (dual of [624, 617, 5]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(253, 624, F25, 2) (dual of [624, 621, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 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
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- OA 2-folding and stacking [i] based on linear OA(257, 628, F25, 4) (dual of [628, 621, 5]-code), using
- net defined by OOA [i] based on linear OOA(257, 314, F25, 4, 4) (dual of [(314, 4), 1249, 5]-NRT-code), using
- digital (16, 25, 3906)-net over F25, using
- net defined by OOA [i] based on linear OOA(2525, 3906, F25, 9, 9) (dual of [(3906, 9), 35129, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2525, 15625, F25, 9) (dual of [15625, 15600, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(2525, 15625, F25, 9) (dual of [15625, 15600, 10]-code), using
- net defined by OOA [i] based on linear OOA(2525, 3906, F25, 9, 9) (dual of [(3906, 9), 35129, 10]-NRT-code), using
- digital (3, 7, 314)-net over F25, using
(23, 23+9, 61273)-Net over F25 — Digital
Digital (23, 32, 61273)-net over F25, using
(23, 23+9, large)-Net in Base 25 — Upper bound on s
There is no (23, 32, large)-net in base 25, because
- 7 times m-reduction [i] would yield (23, 25, large)-net in base 25, but