Best Known (109, 109+26, s)-Nets in Base 4
(109, 109+26, 1260)-Net over F4 — Constructive and digital
Digital (109, 135, 1260)-net over F4, using
- 41 times duplication [i] based on digital (108, 134, 1260)-net over F4, using
- net defined by OOA [i] based on linear OOA(4134, 1260, F4, 26, 26) (dual of [(1260, 26), 32626, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4134, 16380, F4, 26) (dual of [16380, 16246, 27]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4134, 16380, F4, 26) (dual of [16380, 16246, 27]-code), using
- net defined by OOA [i] based on linear OOA(4134, 1260, F4, 26, 26) (dual of [(1260, 26), 32626, 27]-NRT-code), using
(109, 109+26, 8196)-Net over F4 — Digital
Digital (109, 135, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4135, 8196, F4, 2, 26) (dual of [(8196, 2), 16257, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4135, 16392, F4, 26) (dual of [16392, 16257, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4134, 16391, F4, 26) (dual of [16391, 16257, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 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(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4134, 16391, F4, 26) (dual of [16391, 16257, 27]-code), using
- OOA 2-folding [i] based on linear OA(4135, 16392, F4, 26) (dual of [16392, 16257, 27]-code), using
(109, 109+26, 3376331)-Net in Base 4 — Upper bound on s
There is no (109, 135, 3376332)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1897 143393 929919 018928 664850 826864 882675 604228 761786 647932 807666 977446 717263 419120 > 4135 [i]