Best Known (102, 102+21, s)-Nets in Base 5
(102, 102+21, 7818)-Net over F5 — Constructive and digital
Digital (102, 123, 7818)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (92, 113, 7812)-net over F5, using
- net defined by OOA [i] based on linear OOA(5113, 7812, F5, 21, 21) (dual of [(7812, 21), 163939, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5113, 78121, F5, 21) (dual of [78121, 78008, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5113, 78121, F5, 21) (dual of [78121, 78008, 22]-code), using
- net defined by OOA [i] based on linear OOA(5113, 7812, F5, 21, 21) (dual of [(7812, 21), 163939, 22]-NRT-code), using
- digital (0, 10, 6)-net over F5, using
(102, 102+21, 60980)-Net over F5 — Digital
Digital (102, 123, 60980)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 60980, F5, 21) (dual of [60980, 60857, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 78137, F5, 21) (dual of [78137, 78014, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([1,10]) [i] based on
- linear OA(5113, 78126, F5, 21) (dual of [78126, 78013, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(5112, 78126, F5, 10) (dual of [78126, 78014, 11]-code), using the narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(510, 11, F5, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,5)), using
- dual of repetition code with length 11 [i]
- construction X applied to C([0,10]) ⊂ C([1,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 78137, F5, 21) (dual of [78137, 78014, 22]-code), using
(102, 102+21, large)-Net in Base 5 — Upper bound on s
There is no (102, 123, large)-net in base 5, because
- 19 times m-reduction [i] would yield (102, 104, large)-net in base 5, but