Best Known (111, 111+28, s)-Nets in Base 5
(111, 111+28, 1118)-Net over F5 — Constructive and digital
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
(111, 111+28, 13507)-Net over F5 — Digital
Digital (111, 139, 13507)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5139, 13507, F5, 28) (dual of [13507, 13368, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, 15651, F5, 28) (dual of [15651, 15512, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(23) ⊂ 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(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(54, 24, F5, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(27) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5139, 15651, F5, 28) (dual of [15651, 15512, 29]-code), using
(111, 111+28, large)-Net in Base 5 — Upper bound on s
There is no (111, 139, large)-net in base 5, because
- 26 times m-reduction [i] would yield (111, 113, large)-net in base 5, but