Best Known (35, 35+9, s)-Nets in Base 5
(35, 35+9, 3909)-Net over F5 — Constructive and digital
Digital (35, 44, 3909)-net over F5, using
- net defined by OOA [i] based on linear OOA(544, 3909, F5, 9, 9) (dual of [(3909, 9), 35137, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(544, 15637, F5, 9) (dual of [15637, 15593, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(544, 15638, F5, 9) (dual of [15638, 15594, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [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(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 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(544, 15638, F5, 9) (dual of [15638, 15594, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(544, 15637, F5, 9) (dual of [15637, 15593, 10]-code), using
(35, 35+9, 15639)-Net over F5 — Digital
Digital (35, 44, 15639)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(544, 15639, F5, 9) (dual of [15639, 15595, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [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(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(513, 14, F5, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,5)), using
- dual of repetition code with length 14 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 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
(35, 35+9, large)-Net in Base 5 — Upper bound on s
There is no (35, 44, large)-net in base 5, because
- 7 times m-reduction [i] would yield (35, 37, large)-net in base 5, but