Best Known (122, 133, s)-Nets in Base 5
(122, 133, 3630864)-Net over F5 — Constructive and digital
Digital (122, 133, 3630864)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (11, 14, 976572)-net over F5, using
- s-reduction based on digital (11, 14, 1123200)-net over F5, using
- net defined by OOA [i] based on linear OOA(514, 1123200, F5, 3, 3) (dual of [(1123200, 3), 3369586, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(514, 1123200, F5, 2, 3) (dual of [(1123200, 2), 2246386, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(514, 1123200, F5, 3, 3) (dual of [(1123200, 3), 3369586, 4]-NRT-code), using
- s-reduction based on digital (11, 14, 1123200)-net over F5, using
- digital (33, 38, 976572)-net over F5, using
- net defined by OOA [i] based on linear OOA(538, 976572, F5, 5, 5) (dual of [(976572, 5), 4882822, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(538, 1953145, F5, 5) (dual of [1953145, 1953107, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(537, 1953126, F5, 5) (dual of [1953126, 1953089, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(519, 1953126, F5, 3) (dual of [1953126, 1953107, 4]-code or 1953126-cap in PG(18,5)), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(51, 19, F5, 1) (dual of [19, 18, 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 X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(538, 1953145, F5, 5) (dual of [1953145, 1953107, 6]-code), using
- net defined by OOA [i] based on linear OOA(538, 976572, F5, 5, 5) (dual of [(976572, 5), 4882822, 6]-NRT-code), using
- digital (70, 81, 1677720)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- digital (11, 14, 976572)-net over F5, using
(122, 133, large)-Net over F5 — Digital
Digital (122, 133, large)-net over F5, using
- t-expansion [i] based on digital (121, 133, large)-net over F5, using
- 4 times m-reduction [i] based on digital (121, 137, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5137, large, F5, 16) (dual of [large, large−137, 17]-code), using
- 16 times code embedding in larger space [i] based on linear OA(5121, large, F5, 16) (dual of [large, large−121, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 16 times code embedding in larger space [i] based on linear OA(5121, large, F5, 16) (dual of [large, large−121, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5137, large, F5, 16) (dual of [large, large−137, 17]-code), using
- 4 times m-reduction [i] based on digital (121, 137, large)-net over F5, using
(122, 133, large)-Net in Base 5 — Upper bound on s
There is no (122, 133, large)-net in base 5, because
- 9 times m-reduction [i] would yield (122, 124, large)-net in base 5, but