Best Known (116−15, 116, s)-Nets in Base 5
(116−15, 116, 279027)-Net over F5 — Constructive and digital
Digital (101, 116, 279027)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (93, 108, 279017)-net over F5, using
- net defined by OOA [i] based on linear OOA(5108, 279017, F5, 15, 15) (dual of [(279017, 15), 4185147, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5108, 1953120, F5, 15) (dual of [1953120, 1953012, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5108, 1953124, F5, 15) (dual of [1953124, 1953016, 16]-code), using
- 1 times truncation [i] based on linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(5108, 1953124, F5, 15) (dual of [1953124, 1953016, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5108, 1953120, F5, 15) (dual of [1953120, 1953012, 16]-code), using
- net defined by OOA [i] based on linear OOA(5108, 279017, F5, 15, 15) (dual of [(279017, 15), 4185147, 16]-NRT-code), using
- digital (1, 8, 10)-net over F5, using
(116−15, 116, 1953170)-Net over F5 — Digital
Digital (101, 116, 1953170)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5116, 1953170, F5, 15) (dual of [1953170, 1953054, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5114, 1953167, F5, 15) (dual of [1953167, 1953053, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(5109, 1953126, F5, 15) (dual of [1953126, 1953017, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(55, 41, F5, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(5114, 1953168, F5, 13) (dual of [1953168, 1953054, 14]-code), using Gilbert–Varšamov bound and bm = 5114 > Vbs−1(k−1) = 107 954676 278300 620272 304741 516008 420354 493324 202169 626192 695764 307318 492045 [i]
- linear OA(51, 2, F5, 1) (dual of [2, 1, 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(5114, 1953167, F5, 15) (dual of [1953167, 1953053, 16]-code), using
- construction X with Varšamov bound [i] based on
(116−15, 116, large)-Net in Base 5 — Upper bound on s
There is no (101, 116, large)-net in base 5, because
- 13 times m-reduction [i] would yield (101, 103, large)-net in base 5, but