Best Known (32−6, 32, s)-Nets in Base 5
(32−6, 32, 26047)-Net over F5 — Constructive and digital
Digital (26, 32, 26047)-net over F5, using
- net defined by OOA [i] based on linear OOA(532, 26047, F5, 6, 6) (dual of [(26047, 6), 156250, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(532, 78141, F5, 6) (dual of [78141, 78109, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(532, 78142, F5, 6) (dual of [78142, 78110, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(515, 78125, F5, 3) (dual of [78125, 78110, 4]-code or 78125-cap in PG(14,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(53, 17, F5, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(532, 78142, F5, 6) (dual of [78142, 78110, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(532, 78141, F5, 6) (dual of [78141, 78109, 7]-code), using
(32−6, 32, 78142)-Net over F5 — Digital
Digital (26, 32, 78142)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(532, 78142, F5, 6) (dual of [78142, 78110, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(515, 78125, F5, 3) (dual of [78125, 78110, 4]-code or 78125-cap in PG(14,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(53, 17, F5, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(32−6, 32, large)-Net in Base 5 — Upper bound on s
There is no (26, 32, large)-net in base 5, because
- 4 times m-reduction [i] would yield (26, 28, large)-net in base 5, but