Best Known (112, 112+20, s)-Nets in Base 3
(112, 112+20, 5906)-Net over F3 — Constructive and digital
Digital (112, 132, 5906)-net over F3, using
- net defined by OOA [i] based on linear OOA(3132, 5906, F3, 20, 20) (dual of [(5906, 20), 117988, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3132, 59060, F3, 20) (dual of [59060, 58928, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3132, 59060, F3, 20) (dual of [59060, 58928, 21]-code), using
(112, 112+20, 19686)-Net over F3 — Digital
Digital (112, 132, 19686)-net over F3, using
- 31 times duplication [i] based on digital (111, 131, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3131, 19686, F3, 3, 20) (dual of [(19686, 3), 58927, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3131, 59058, F3, 20) (dual of [59058, 58927, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- OOA 3-folding [i] based on linear OA(3131, 59058, F3, 20) (dual of [59058, 58927, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3131, 19686, F3, 3, 20) (dual of [(19686, 3), 58927, 21]-NRT-code), using
(112, 112+20, 4497238)-Net in Base 3 — Upper bound on s
There is no (112, 132, 4497239)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 955 005295 736926 488483 347581 624877 318477 452398 552162 581672 726557 > 3132 [i]