Best Known (120, 120+29, s)-Nets in Base 4
(120, 120+29, 1170)-Net over F4 — Constructive and digital
Digital (120, 149, 1170)-net over F4, using
- 41 times duplication [i] based on digital (119, 148, 1170)-net over F4, using
- net defined by OOA [i] based on linear OOA(4148, 1170, F4, 29, 29) (dual of [(1170, 29), 33782, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4148, 16381, F4, 29) (dual of [16381, 16233, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4148, 16381, F4, 29) (dual of [16381, 16233, 30]-code), using
- net defined by OOA [i] based on linear OOA(4148, 1170, F4, 29, 29) (dual of [(1170, 29), 33782, 30]-NRT-code), using
(120, 120+29, 8196)-Net over F4 — Digital
Digital (120, 149, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4149, 8196, F4, 2, 29) (dual of [(8196, 2), 16243, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4149, 16392, F4, 29) (dual of [16392, 16243, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(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(28) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(4149, 16392, F4, 29) (dual of [16392, 16243, 30]-code), using
(120, 120+29, 4666267)-Net in Base 4 — Upper bound on s
There is no (120, 149, 4666268)-net in base 4, because
- 1 times m-reduction [i] would yield (120, 148, 4666268)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 127314 907579 464890 619284 318594 410351 496824 673263 888612 501722 753781 137429 417424 019791 966300 > 4148 [i]