Best Known (15, 15+4, s)-Nets in Base 5
(15, 15+4, 7815)-Net over F5 — Constructive and digital
Digital (15, 19, 7815)-net over F5, using
- net defined by OOA [i] based on linear OOA(519, 7815, F5, 4, 4) (dual of [(7815, 4), 31241, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(519, 7815, F5, 3, 4) (dual of [(7815, 3), 23426, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(519, 15630, F5, 4) (dual of [15630, 15611, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(519, 15631, F5, 4) (dual of [15631, 15612, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(513, 15625, F5, 3) (dual of [15625, 15612, 4]-code or 15625-cap in PG(12,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(519, 15631, F5, 4) (dual of [15631, 15612, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(519, 15630, F5, 4) (dual of [15630, 15611, 5]-code), using
- appending kth column [i] based on linear OOA(519, 7815, F5, 3, 4) (dual of [(7815, 3), 23426, 5]-NRT-code), using
(15, 15+4, 15631)-Net over F5 — Digital
Digital (15, 19, 15631)-net over F5, using
- net defined by OOA [i] based on linear OOA(519, 15631, F5, 4, 4) (dual of [(15631, 4), 62505, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(519, 15631, F5, 3, 4) (dual of [(15631, 3), 46874, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(519, 15631, F5, 4) (dual of [15631, 15612, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(513, 15625, F5, 3) (dual of [15625, 15612, 4]-code or 15625-cap in PG(12,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(519, 15631, F5, 4) (dual of [15631, 15612, 5]-code), using
- appending kth column [i] based on linear OOA(519, 15631, F5, 3, 4) (dual of [(15631, 3), 46874, 5]-NRT-code), using
(15, 15+4, 1544079)-Net in Base 5 — Upper bound on s
There is no (15, 19, 1544080)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 19 073489 076481 > 519 [i]