Best Known (138−18, 138, s)-Nets in Base 5
(138−18, 138, 217026)-Net over F5 — Constructive and digital
Digital (120, 138, 217026)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (109, 127, 217014)-net over F5, using
- net defined by OOA [i] based on linear OOA(5127, 217014, F5, 18, 18) (dual of [(217014, 18), 3906125, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5127, 1953126, F5, 18) (dual of [1953126, 1952999, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 1953134, F5, 18) (dual of [1953134, 1953007, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5127, 1953134, F5, 18) (dual of [1953134, 1953007, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5127, 1953126, F5, 18) (dual of [1953126, 1952999, 19]-code), using
- net defined by OOA [i] based on linear OOA(5127, 217014, F5, 18, 18) (dual of [(217014, 18), 3906125, 19]-NRT-code), using
- digital (2, 11, 12)-net over F5, using
(138−18, 138, 1642126)-Net over F5 — Digital
Digital (120, 138, 1642126)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 1642126, F5, 18) (dual of [1642126, 1641988, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5138, 1953139, F5, 18) (dual of [1953139, 1953001, 19]-code), using
- (u, u+v)-construction [i] based on
- linear OA(511, 14, F5, 9) (dual of [14, 3, 10]-code), using
- 2 times truncation [i] based on linear OA(513, 16, F5, 11) (dual of [16, 3, 12]-code), using
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(511, 14, F5, 9) (dual of [14, 3, 10]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5138, 1953139, F5, 18) (dual of [1953139, 1953001, 19]-code), using
(138−18, 138, large)-Net in Base 5 — Upper bound on s
There is no (120, 138, large)-net in base 5, because
- 16 times m-reduction [i] would yield (120, 122, large)-net in base 5, but