Best Known (107−21, 107, s)-Nets in Base 4
(107−21, 107, 1639)-Net over F4 — Constructive and digital
Digital (86, 107, 1639)-net over F4, using
- net defined by OOA [i] based on linear OOA(4107, 1639, F4, 21, 21) (dual of [(1639, 21), 34312, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4107, 16391, F4, 21) (dual of [16391, 16284, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4107, 16391, F4, 21) (dual of [16391, 16284, 22]-code), using
(107−21, 107, 8171)-Net over F4 — Digital
Digital (86, 107, 8171)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4107, 8171, F4, 2, 21) (dual of [(8171, 2), 16235, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4107, 8196, F4, 2, 21) (dual of [(8196, 2), 16285, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(4107, 16392, F4, 21) (dual of [16392, 16285, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(4107, 8196, F4, 2, 21) (dual of [(8196, 2), 16285, 22]-NRT-code), using
(107−21, 107, 3636553)-Net in Base 4 — Upper bound on s
There is no (86, 107, 3636554)-net in base 4, because
- 1 times m-reduction [i] would yield (86, 106, 3636554)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6582 021220 753666 663557 418777 449214 356821 905942 517345 459121 676456 > 4106 [i]