Best Known (13, 13+6, s)-Nets in Base 5
(13, 13+6, 210)-Net over F5 — Constructive and digital
Digital (13, 19, 210)-net over F5, using
- 51 times duplication [i] based on digital (12, 18, 210)-net over F5, using
- net defined by OOA [i] based on linear OOA(518, 210, F5, 6, 6) (dual of [(210, 6), 1242, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(517, 625, F5, 6) (dual of [625, 608, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(513, 625, F5, 4) (dual of [625, 612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 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(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
- net defined by OOA [i] based on linear OOA(518, 210, F5, 6, 6) (dual of [(210, 6), 1242, 7]-NRT-code), using
(13, 13+6, 633)-Net over F5 — Digital
Digital (13, 19, 633)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(519, 633, F5, 6) (dual of [633, 614, 7]-code), using
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(517, 625, F5, 6) (dual of [625, 608, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(513, 625, F5, 4) (dual of [625, 612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 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(5) ⊂ Ce(3) [i] based on
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
(13, 13+6, 12135)-Net in Base 5 — Upper bound on s
There is no (13, 19, 12136)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 19 074065 811105 > 519 [i]