Best Known (140−28, 140, s)-Nets in Base 5
(140−28, 140, 1118)-Net over F5 — Constructive and digital
Digital (112, 140, 1118)-net over F5, using
- 51 times duplication [i] based on digital (111, 139, 1118)-net over F5, using
- net defined by OOA [i] based on linear OOA(5139, 1118, F5, 28, 28) (dual of [(1118, 28), 31165, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(5139, 15652, F5, 28) (dual of [15652, 15513, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, 15655, F5, 28) (dual of [15655, 15516, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5139, 15655, F5, 28) (dual of [15655, 15516, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(5139, 15652, F5, 28) (dual of [15652, 15513, 29]-code), using
- net defined by OOA [i] based on linear OOA(5139, 1118, F5, 28, 28) (dual of [(1118, 28), 31165, 29]-NRT-code), using
(140−28, 140, 14371)-Net over F5 — Digital
Digital (112, 140, 14371)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5140, 14371, F5, 28) (dual of [14371, 14231, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5140, 15651, F5, 28) (dual of [15651, 15511, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(57, 26, F5, 4) (dual of [26, 19, 5]-code), using
- base reduction for projective spaces (embedding PG(3,25) in PG(6,5)) [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- extended Reed–Solomon code RSe(22,25) [i]
- algebraic-geometric code AG(F, Q+9P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,7P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- base reduction for projective spaces (embedding PG(3,25) in PG(6,5)) [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5140, 15651, F5, 28) (dual of [15651, 15511, 29]-code), using
(140−28, 140, large)-Net in Base 5 — Upper bound on s
There is no (112, 140, large)-net in base 5, because
- 26 times m-reduction [i] would yield (112, 114, large)-net in base 5, but