Best Known (43−9, 43, s)-Nets in Base 5
(43−9, 43, 3907)-Net over F5 — Constructive and digital
Digital (34, 43, 3907)-net over F5, using
- net defined by OOA [i] based on linear OOA(543, 3907, F5, 9, 9) (dual of [(3907, 9), 35120, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(543, 15629, F5, 9) (dual of [15629, 15586, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(543, 15631, F5, 9) (dual of [15631, 15588, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(543, 15631, F5, 9) (dual of [15631, 15588, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(543, 15629, F5, 9) (dual of [15629, 15586, 10]-code), using
(43−9, 43, 13199)-Net over F5 — Digital
Digital (34, 43, 13199)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(543, 13199, F5, 9) (dual of [13199, 13156, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using
(43−9, 43, large)-Net in Base 5 — Upper bound on s
There is no (34, 43, large)-net in base 5, because
- 7 times m-reduction [i] would yield (34, 36, large)-net in base 5, but