Best Known (92, 92+8, s)-Nets in Base 5
(92, 92+8, 5170867)-Net over F5 — Constructive and digital
Digital (92, 100, 5170867)-net over F5, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(528, 976567, F5, 4, 4) (dual of [(976567, 4), 3906240, 5]-NRT-code), using
- digital (64, 72, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 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
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- digital (24, 28, 976567)-net over F5, using
(92, 92+8, large)-Net over F5 — Digital
Digital (92, 100, large)-net over F5, using
- t-expansion [i] based on digital (88, 100, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5100, large, F5, 12) (dual of [large, large−100, 13]-code), using
- 9 times code embedding in larger space [i] based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 9 times code embedding in larger space [i] based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5100, large, F5, 12) (dual of [large, large−100, 13]-code), using
(92, 92+8, large)-Net in Base 5 — Upper bound on s
There is no (92, 100, large)-net in base 5, because
- 6 times m-reduction [i] would yield (92, 94, large)-net in base 5, but