Best Known (113−28, 113, s)-Nets in Base 5
(113−28, 113, 400)-Net over F5 — Constructive and digital
Digital (85, 113, 400)-net over F5, using
- 7 times m-reduction [i] based on digital (85, 120, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 60, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 60, 200)-net over F25, using
(113−28, 113, 2687)-Net over F5 — Digital
Digital (85, 113, 2687)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5113, 2687, F5, 28) (dual of [2687, 2574, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 3137, F5, 28) (dual of [3137, 3024, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5112, 3136, F5, 28) (dual of [3136, 3024, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(5111, 3125, F5, 28) (dual of [3125, 3014, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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(27) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5112, 3136, F5, 28) (dual of [3136, 3024, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 3137, F5, 28) (dual of [3137, 3024, 29]-code), using
(113−28, 113, 662334)-Net in Base 5 — Upper bound on s
There is no (85, 113, 662335)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 9 629665 395056 857733 829949 460794 954674 900729 525083 486559 781421 816933 009536 870345 > 5113 [i]