Best Known (49, 58, s)-Nets in Base 5
(49, 58, 97660)-Net over F5 — Constructive and digital
Digital (49, 58, 97660)-net over F5, using
- net defined by OOA [i] based on linear OOA(558, 97660, F5, 9, 9) (dual of [(97660, 9), 878882, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(558, 390641, F5, 9) (dual of [390641, 390583, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(558, 390642, F5, 9) (dual of [390642, 390584, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(51, 17, F5, 1) (dual of [17, 16, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(558, 390642, F5, 9) (dual of [390642, 390584, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(558, 390641, F5, 9) (dual of [390641, 390583, 10]-code), using
(49, 58, 390643)-Net over F5 — Digital
Digital (49, 58, 390643)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(558, 390643, F5, 9) (dual of [390643, 390585, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 18, F5, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,5)), using
- dual of repetition code with length 18 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
(49, 58, large)-Net in Base 5 — Upper bound on s
There is no (49, 58, large)-net in base 5, because
- 7 times m-reduction [i] would yield (49, 51, large)-net in base 5, but