Best Known (27, 44, s)-Nets in Base 5
(27, 44, 132)-Net over F5 — Constructive and digital
Digital (27, 44, 132)-net over F5, using
- 2 times m-reduction [i] based on digital (27, 46, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 23, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 23, 66)-net over F25, using
(27, 44, 151)-Net over F5 — Digital
Digital (27, 44, 151)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(544, 151, F5, 17) (dual of [151, 107, 18]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 9 times 0) [i] based on linear OA(540, 128, F5, 17) (dual of [128, 88, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(540, 125, F5, 17) (dual of [125, 85, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(537, 125, F5, 16) (dual of [125, 88, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 9 times 0) [i] based on linear OA(540, 128, F5, 17) (dual of [128, 88, 18]-code), using
(27, 44, 5372)-Net in Base 5 — Upper bound on s
There is no (27, 44, 5373)-net in base 5, because
- 1 times m-reduction [i] would yield (27, 43, 5373)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 138275 596992 471420 588813 596385 > 543 [i]