Best Known (29−8, 29, s)-Nets in Base 5
(29−8, 29, 164)-Net over F5 — Constructive and digital
Digital (21, 29, 164)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (17, 25, 158)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 158, F5, 8, 8) (dual of [(158, 8), 1239, 9]-NRT-code), using
- OA 4-folding and stacking [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
- OA 4-folding and stacking [i] based on linear OA(525, 632, F5, 8) (dual of [632, 607, 9]-code), using
- net defined by OOA [i] based on linear OOA(525, 158, F5, 8, 8) (dual of [(158, 8), 1239, 9]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(29−8, 29, 726)-Net over F5 — Digital
Digital (21, 29, 726)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(529, 726, F5, 8) (dual of [726, 697, 9]-code), using
- 90 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 19 times 0, 1, 65 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
- 90 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 19 times 0, 1, 65 times 0) [i] based on linear OA(525, 632, F5, 8) (dual of [632, 607, 9]-code), using
(29−8, 29, 64641)-Net in Base 5 — Upper bound on s
There is no (21, 29, 64642)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 186 275203 010072 865345 > 529 [i]