Best Known (144, 144+24, s)-Nets in Base 3
(144, 144+24, 4923)-Net over F3 — Constructive and digital
Digital (144, 168, 4923)-net over F3, using
- t-expansion [i] based on digital (143, 168, 4923)-net over F3, using
- net defined by OOA [i] based on linear OOA(3168, 4923, F3, 25, 25) (dual of [(4923, 25), 122907, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3168, 59077, F3, 25) (dual of [59077, 58909, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3167, 59076, F3, 25) (dual of [59076, 58909, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3141, 59050, F3, 21) (dual of [59050, 58909, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3167, 59076, F3, 25) (dual of [59076, 58909, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3168, 59077, F3, 25) (dual of [59077, 58909, 26]-code), using
- net defined by OOA [i] based on linear OOA(3168, 4923, F3, 25, 25) (dual of [(4923, 25), 122907, 26]-NRT-code), using
(144, 144+24, 25624)-Net over F3 — Digital
Digital (144, 168, 25624)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3168, 25624, F3, 2, 24) (dual of [(25624, 2), 51080, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3168, 29543, F3, 2, 24) (dual of [(29543, 2), 58918, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3168, 59086, F3, 24) (dual of [59086, 58918, 25]-code), using
- 1 times truncation [i] based on linear OA(3169, 59087, F3, 25) (dual of [59087, 58918, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(3169, 59087, F3, 25) (dual of [59087, 58918, 26]-code), using
- OOA 2-folding [i] based on linear OA(3168, 59086, F3, 24) (dual of [59086, 58918, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3168, 29543, F3, 2, 24) (dual of [(29543, 2), 58918, 25]-NRT-code), using
(144, 144+24, large)-Net in Base 3 — Upper bound on s
There is no (144, 168, large)-net in base 3, because
- 22 times m-reduction [i] would yield (144, 146, large)-net in base 3, but