Best Known (141−27, 141, s)-Nets in Base 4
(141−27, 141, 1260)-Net over F4 — Constructive and digital
Digital (114, 141, 1260)-net over F4, using
- net defined by OOA [i] based on linear OOA(4141, 1260, F4, 27, 27) (dual of [(1260, 27), 33879, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4141, 16381, F4, 27) (dual of [16381, 16240, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4141, 16381, F4, 27) (dual of [16381, 16240, 28]-code), using
(141−27, 141, 8195)-Net over F4 — Digital
Digital (114, 141, 8195)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4141, 8195, F4, 2, 27) (dual of [(8195, 2), 16249, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4141, 16390, F4, 27) (dual of [16390, 16249, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 16391, F4, 27) (dual of [16391, 16250, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4141, 16391, F4, 27) (dual of [16391, 16250, 28]-code), using
- OOA 2-folding [i] based on linear OA(4141, 16390, F4, 27) (dual of [16390, 16249, 28]-code), using
(141−27, 141, 5754493)-Net in Base 4 — Upper bound on s
There is no (114, 141, 5754494)-net in base 4, because
- 1 times m-reduction [i] would yield (114, 140, 5754494)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 942669 731995 627580 190170 413389 156583 650396 632984 967567 656266 033513 430611 655105 107425 > 4140 [i]