Best Known (110, 110+28, s)-Nets in Base 5
(110, 110+28, 1117)-Net over F5 — Constructive and digital
Digital (110, 138, 1117)-net over F5, using
- 54 times duplication [i] based on digital (106, 134, 1117)-net over F5, using
- net defined by OOA [i] based on linear OOA(5134, 1117, F5, 28, 28) (dual of [(1117, 28), 31142, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(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(51, 13, F5, 1) (dual of [13, 12, 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
- OA 14-folding and stacking [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- net defined by OOA [i] based on linear OOA(5134, 1117, F5, 28, 28) (dual of [(1117, 28), 31142, 29]-NRT-code), using
(110, 110+28, 12695)-Net over F5 — Digital
Digital (110, 138, 12695)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 12695, F5, 28) (dual of [12695, 12557, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5138, 15649, F5, 28) (dual of [15649, 15511, 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, 23, F5, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(27) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5138, 15649, F5, 28) (dual of [15649, 15511, 29]-code), using
(110, 110+28, large)-Net in Base 5 — Upper bound on s
There is no (110, 138, large)-net in base 5, because
- 26 times m-reduction [i] would yield (110, 112, large)-net in base 5, but