Best Known (32−8, 32, s)-Nets in Base 5
(32−8, 32, 784)-Net over F5 — Constructive and digital
Digital (24, 32, 784)-net over F5, using
- net defined by OOA [i] based on linear OOA(532, 784, F5, 8, 8) (dual of [(784, 8), 6240, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(532, 3136, F5, 8) (dual of [3136, 3104, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(532, 3136, F5, 8) (dual of [3136, 3104, 9]-code), using
(32−8, 32, 3055)-Net over F5 — Digital
Digital (24, 32, 3055)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(532, 3055, F5, 8) (dual of [3055, 3023, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(532, 3136, F5, 8) (dual of [3136, 3104, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(531, 3125, F5, 8) (dual of [3125, 3094, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(532, 3136, F5, 8) (dual of [3136, 3104, 9]-code), using
(32−8, 32, 216146)-Net in Base 5 — Upper bound on s
There is no (24, 32, 216147)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 23283 360610 856705 955185 > 532 [i]