Best Known (20, 20+5, s)-Nets in Base 5
(20, 20+5, 7815)-Net over F5 — Constructive and digital
Digital (20, 25, 7815)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 7815, F5, 5, 5) (dual of [(7815, 5), 39050, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(525, 15631, F5, 5) (dual of [15631, 15606, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(525, 15632, F5, 5) (dual of [15632, 15607, 6]-code), using
- 1 times truncation [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(57, 8, F5, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,5)), using
- dual of repetition code with length 8 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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(5) ⊂ Ce(3) [i] based on
- 1 times truncation [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(525, 15632, F5, 5) (dual of [15632, 15607, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(525, 15631, F5, 5) (dual of [15631, 15606, 6]-code), using
(20, 20+5, 15632)-Net over F5 — Digital
Digital (20, 25, 15632)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(525, 15632, F5, 5) (dual of [15632, 15607, 6]-code), using
- 1 times truncation [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(57, 8, F5, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,5)), using
- dual of repetition code with length 8 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 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(5) ⊂ Ce(3) [i] based on
- 1 times truncation [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
(20, 20+5, large)-Net in Base 5 — Upper bound on s
There is no (20, 25, large)-net in base 5, because
- 3 times m-reduction [i] would yield (20, 22, large)-net in base 5, but