Best Known (113, 113+20, s)-Nets in Base 5
(113, 113+20, 39065)-Net over F5 — Constructive and digital
Digital (113, 133, 39065)-net over F5, using
- t-expansion [i] based on digital (112, 133, 39065)-net over F5, using
- net defined by OOA [i] based on linear OOA(5133, 39065, F5, 21, 21) (dual of [(39065, 21), 820232, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5133, 390651, F5, 21) (dual of [390651, 390518, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(5133, 390651, F5, 21) (dual of [390651, 390518, 22]-code), using
- net defined by OOA [i] based on linear OOA(5133, 39065, F5, 21, 21) (dual of [(39065, 21), 820232, 22]-NRT-code), using
(113, 113+20, 252248)-Net over F5 — Digital
Digital (113, 133, 252248)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5133, 252248, F5, 20) (dual of [252248, 252115, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 390654, F5, 20) (dual of [390654, 390521, 21]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(53, 28, F5, 2) (dual of [28, 25, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), 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(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5133, 390654, F5, 20) (dual of [390654, 390521, 21]-code), using
(113, 113+20, large)-Net in Base 5 — Upper bound on s
There is no (113, 133, large)-net in base 5, because
- 18 times m-reduction [i] would yield (113, 115, large)-net in base 5, but