Best Known (20, 20+8, s)-Nets in Base 5
(20, 20+8, 159)-Net over F5 — Constructive and digital
Digital (20, 28, 159)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(527, 159, F5, 8, 8) (dual of [(159, 8), 1245, 9]-NRT-code), using
(20, 20+8, 659)-Net over F5 — Digital
Digital (20, 28, 659)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(528, 659, F5, 8) (dual of [659, 631, 9]-code), using
- 24 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 19 times 0) [i] based on linear OA(525, 632, F5, 8) (dual of [632, 607, 9]-code), using
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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 = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [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(50, 4, F5, 0) (dual of [4, 4, 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
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- 24 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 19 times 0) [i] based on linear OA(525, 632, F5, 8) (dual of [632, 607, 9]-code), using
(20, 20+8, 43227)-Net in Base 5 — Upper bound on s
There is no (20, 28, 43228)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 37 255445 029650 964225 > 528 [i]