Best Known (135−26, 135, s)-Nets in Base 5
(135−26, 135, 1211)-Net over F5 — Constructive and digital
Digital (109, 135, 1211)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 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 (95, 121, 1201)-net over F5, using
- net defined by OOA [i] based on linear OOA(5121, 1201, F5, 26, 26) (dual of [(1201, 26), 31105, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(5121, 15613, F5, 26) (dual of [15613, 15492, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(5121, 15613, F5, 26) (dual of [15613, 15492, 27]-code), using
- net defined by OOA [i] based on linear OOA(5121, 1201, F5, 26, 26) (dual of [(1201, 26), 31105, 27]-NRT-code), using
- digital (1, 14, 10)-net over F5, using
(135−26, 135, 15676)-Net over F5 — Digital
Digital (109, 135, 15676)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5135, 15676, F5, 26) (dual of [15676, 15541, 27]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5134, 15674, F5, 26) (dual of [15674, 15540, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(513, 49, F5, 7) (dual of [49, 36, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(513, 52, F5, 7) (dual of [52, 39, 8]-code), using
- a “LX†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(513, 52, F5, 7) (dual of [52, 39, 8]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(5134, 15675, F5, 25) (dual of [15675, 15541, 26]-code), using Gilbert–Varšamov bound and bm = 5134 > Vbs−1(k−1) = 21 555874 235072 028556 907080 021809 290092 783795 512952 238809 345112 329278 057487 273207 155042 055225 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(5134, 15674, F5, 26) (dual of [15674, 15540, 27]-code), using
- construction X with Varšamov bound [i] based on
(135−26, 135, large)-Net in Base 5 — Upper bound on s
There is no (109, 135, large)-net in base 5, because
- 24 times m-reduction [i] would yield (109, 111, large)-net in base 5, but