Best Known (19, 19+8, s)-Nets in Base 5
(19, 19+8, 159)-Net over F5 — Constructive and digital
Digital (19, 27, 159)-net over F5, using
- net defined by OOA [i] based on linear OOA(527, 159, F5, 8, 8) (dual of [(159, 8), 1245, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(527, 636, F5, 8) (dual of [636, 609, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(527, 638, F5, 8) (dual of [638, 611, 9]-code), using
- construction XX applied to C1 = C([622,3]), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([622,5]) [i] based on
- linear OA(521, 624, F5, 6) (dual of [624, 603, 7]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,3}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(525, 624, F5, 8) (dual of [624, 599, 9]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,5}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(513, 624, F5, 4) (dual of [624, 611, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- construction XX applied to C1 = C([622,3]), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([622,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(527, 638, F5, 8) (dual of [638, 611, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(527, 636, F5, 8) (dual of [636, 609, 9]-code), using
(19, 19+8, 640)-Net over F5 — Digital
Digital (19, 27, 640)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(527, 640, F5, 8) (dual of [640, 613, 9]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(526, 635, F5, 8) (dual of [635, 609, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(525, 625, F5, 8) (dual of [625, 600, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(59, 10, F5, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
- dual of repetition code with length 10 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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(7) ⊂ Ce(5) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(526, 635, F5, 8) (dual of [635, 609, 9]-code), using
(19, 19+8, 28906)-Net in Base 5 — Upper bound on s
There is no (19, 27, 28907)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 7 450596 434452 913265 > 527 [i]