Best Known (133−24, 133, s)-Nets in Base 4
(133−24, 133, 1367)-Net over F4 — Constructive and digital
Digital (109, 133, 1367)-net over F4, using
- 41 times duplication [i] based on digital (108, 132, 1367)-net over F4, using
- t-expansion [i] based on digital (107, 132, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(4132, 1367, F4, 25, 25) (dual of [(1367, 25), 34043, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4132, 16405, F4, 25) (dual of [16405, 16273, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 16410, F4, 25) (dual of [16410, 16278, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- 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(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(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4132, 16410, F4, 25) (dual of [16410, 16278, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4132, 16405, F4, 25) (dual of [16405, 16273, 26]-code), using
- net defined by OOA [i] based on linear OOA(4132, 1367, F4, 25, 25) (dual of [(1367, 25), 34043, 26]-NRT-code), using
- t-expansion [i] based on digital (107, 132, 1367)-net over F4, using
(133−24, 133, 12345)-Net over F4 — Digital
Digital (109, 133, 12345)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4133, 12345, F4, 24) (dual of [12345, 12212, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, 16406, F4, 24) (dual of [16406, 16273, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(46, 21, F4, 4) (dual of [21, 15, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4133, 16406, F4, 24) (dual of [16406, 16273, 25]-code), using
(133−24, 133, 8299868)-Net in Base 4 — Upper bound on s
There is no (109, 133, 8299869)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 118 571117 006908 213336 569882 047653 546441 207524 745663 937649 537189 993254 838648 841555 > 4133 [i]