Best Known (150−29, 150, s)-Nets in Base 4
(150−29, 150, 1170)-Net over F4 — Constructive and digital
Digital (121, 150, 1170)-net over F4, using
- 42 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
(150−29, 150, 8197)-Net over F4 — Digital
Digital (121, 150, 8197)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4150, 8197, F4, 2, 29) (dual of [(8197, 2), 16244, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4150, 16394, F4, 29) (dual of [16394, 16244, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [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(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(41, 9, F4, 1) (dual of [9, 8, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(4150, 16394, F4, 29) (dual of [16394, 16244, 30]-code), using
(150−29, 150, 5151978)-Net in Base 4 — Upper bound on s
There is no (121, 150, 5151979)-net in base 4, because
- 1 times m-reduction [i] would yield (121, 149, 5151979)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 509260 005174 619435 676918 710431 097037 603778 961217 709750 992475 418071 818338 178224 534023 056104 > 4149 [i]