Best Known (12−6, 12, s)-Nets in Base 25
(12−6, 12, 210)-Net over F25 — Constructive and digital
Digital (6, 12, 210)-net over F25, using
- net defined by OOA [i] based on linear OOA(2512, 210, F25, 6, 6) (dual of [(210, 6), 1248, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2512, 630, F25, 6) (dual of [630, 618, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(2511, 625, F25, 6) (dual of [625, 614, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(257, 625, F25, 4) (dual of [625, 618, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(251, 5, F25, 1) (dual of [5, 4, 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
- OA 3-folding and stacking [i] based on linear OA(2512, 630, F25, 6) (dual of [630, 618, 7]-code), using
(12−6, 12, 631)-Net over F25 — Digital
Digital (6, 12, 631)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2512, 631, F25, 6) (dual of [631, 619, 7]-code), using
- construction XX applied to C1 = C([622,2]), C2 = C([0,3]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C([622,3]) [i] based on
- linear OA(259, 624, F25, 5) (dual of [624, 615, 6]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−2,−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(257, 624, F25, 4) (dual of [624, 617, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−2,−1,…,3}, and designed minimum distance d ≥ |I|+1 = 7 [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(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- Reed–Solomon code RS(24,25) [i]
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- 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
- construction XX applied to C1 = C([622,2]), C2 = C([0,3]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C([622,3]) [i] based on
(12−6, 12, 29574)-Net in Base 25 — Upper bound on s
There is no (6, 12, 29575)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 59608 239212 503801 > 2512 [i]