Best Known (46−11, 46, s)-Nets in Base 5
(46−11, 46, 631)-Net over F5 — Constructive and digital
Digital (35, 46, 631)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (30, 41, 625)-net over F5, using
- net defined by OOA [i] based on linear OOA(541, 625, F5, 11, 11) (dual of [(625, 11), 6834, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- net defined by OOA [i] based on linear OOA(541, 625, F5, 11, 11) (dual of [(625, 11), 6834, 12]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(46−11, 46, 3163)-Net over F5 — Digital
Digital (35, 46, 3163)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(546, 3163, F5, 11) (dual of [3163, 3117, 12]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 17 times 0) [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 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 Ce(10) ⊂ Ce(8) [i] based on
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 17 times 0) [i] based on linear OA(542, 3131, F5, 11) (dual of [3131, 3089, 12]-code), using
(46−11, 46, 1272052)-Net in Base 5 — Upper bound on s
There is no (35, 46, 1272053)-net in base 5, because
- 1 times m-reduction [i] would yield (35, 45, 1272053)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 28 421715 719645 270388 665623 765765 > 545 [i]