Best Known (105, 105+21, s)-Nets in Base 5
(105, 105+21, 7828)-Net over F5 — Constructive and digital
Digital (105, 126, 7828)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-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 (3, 13, 16)-net over F5, using
(105, 105+21, 78181)-Net over F5 — Digital
Digital (105, 126, 78181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5126, 78181, F5, 21) (dual of [78181, 78055, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [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(571, 78126, F5, 13) (dual of [78126, 78055, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(513, 55, F5, 7) (dual of [55, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(513, 63, F5, 7) (dual of [63, 50, 8]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(513, 63, F5, 7) (dual of [63, 50, 8]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
(105, 105+21, large)-Net in Base 5 — Upper bound on s
There is no (105, 126, large)-net in base 5, because
- 19 times m-reduction [i] would yield (105, 107, large)-net in base 5, but