Best Known (28−4, 28, s)-Nets in Base 5
(28−4, 28, 976567)-Net over F5 — Constructive and digital
Digital (24, 28, 976567)-net over F5, using
- net defined by OOA [i] based on linear OOA(528, 976567, F5, 4, 4) (dual of [(976567, 4), 3906240, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(528, 976567, F5, 3, 4) (dual of [(976567, 3), 2929673, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(528, 1953134, F5, 4) (dual of [1953134, 1953106, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(519, 1953125, F5, 3) (dual of [1953125, 1953106, 4]-code or 1953125-cap in PG(18,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 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(528, 1953134, F5, 4) (dual of [1953134, 1953106, 5]-code), using
- appending kth column [i] based on linear OOA(528, 976567, F5, 3, 4) (dual of [(976567, 3), 2929673, 5]-NRT-code), using
(28−4, 28, 1953134)-Net over F5 — Digital
Digital (24, 28, 1953134)-net over F5, using
- net defined by OOA [i] based on linear OOA(528, 1953134, F5, 4, 4) (dual of [(1953134, 4), 7812508, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(528, 1953134, F5, 3, 4) (dual of [(1953134, 3), 5859374, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(528, 1953134, F5, 4) (dual of [1953134, 1953106, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(528, 1953125, F5, 4) (dual of [1953125, 1953097, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(519, 1953125, F5, 3) (dual of [1953125, 1953106, 4]-code or 1953125-cap in PG(18,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(528, 1953134, F5, 4) (dual of [1953134, 1953106, 5]-code), using
- appending kth column [i] based on linear OOA(528, 1953134, F5, 3, 4) (dual of [(1953134, 3), 5859374, 5]-NRT-code), using
(28−4, 28, large)-Net in Base 5 — Upper bound on s
There is no (24, 28, large)-net in base 5, because
- 2 times m-reduction [i] would yield (24, 26, large)-net in base 5, but