Best Known (114, 114+29, s)-Nets in Base 5
(114, 114+29, 1117)-Net over F5 — Constructive and digital
Digital (114, 143, 1117)-net over F5, using
- 52 times duplication [i] based on digital (112, 141, 1117)-net over F5, using
- net defined by OOA [i] based on linear OOA(5141, 1117, F5, 29, 29) (dual of [(1117, 29), 32252, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5141, 15639, F5, 29) (dual of [15639, 15498, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5141, 15640, F5, 29) (dual of [15640, 15499, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(5127, 15625, F5, 27) (dual of [15625, 15498, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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, 14, F5, 1) (dual of [14, 13, 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
- 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(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(5141, 15640, F5, 29) (dual of [15640, 15499, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5141, 15639, F5, 29) (dual of [15639, 15498, 30]-code), using
- net defined by OOA [i] based on linear OOA(5141, 1117, F5, 29, 29) (dual of [(1117, 29), 32252, 30]-NRT-code), using
(114, 114+29, 12935)-Net over F5 — Digital
Digital (114, 143, 12935)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5143, 12935, F5, 29) (dual of [12935, 12792, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5143, 15647, F5, 29) (dual of [15647, 15504, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5142, 15646, F5, 29) (dual of [15646, 15504, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(53, 21, F5, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5142, 15646, F5, 29) (dual of [15646, 15504, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5143, 15647, F5, 29) (dual of [15647, 15504, 30]-code), using
(114, 114+29, large)-Net in Base 5 — Upper bound on s
There is no (114, 143, large)-net in base 5, because
- 27 times m-reduction [i] would yield (114, 116, large)-net in base 5, but