MinT - Best Known (151−22, 151, s)-Nets in Base 3

Best Known (151−22, 151, s)-Nets in Base 3

(151−22, 151, 5371)-Net over F₃ — Constructive and digital

Digital (129, 151, 5371)-net over F₃, using

3² times duplication [i] based on digital (127, 149, 5371)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁴⁹, 5371, F₃, 22, 22) (dual of [(5371, 22), 118013, 23]-NRT-code), using
  - OA 11-folding and stacking [i] based on linear OA(3¹⁴⁹, 59081, F₃, 22) (dual of [59081, 58932, 23]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁴⁹, 59087, F₃, 22) (dual of [59087, 58938, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
        linear OA(3¹⁴¹, 59049, F₃, 22) (dual of [59049, 58908, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(3¹¹¹, 59049, F₃, 17) (dual of [59049, 58938, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(151−22, 151, 21851)-Net over F₃ — Digital

Digital (129, 151, 21851)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵¹, 21851, F₃, 2, 22) (dual of [(21851, 2), 43551, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵¹, 29544, F₃, 2, 22) (dual of [(29544, 2), 58937, 23]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁵⁰, 29544, F₃, 2, 22) (dual of [(29544, 2), 58938, 23]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁵⁰, 59088, F₃, 22) (dual of [59088, 58938, 23]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹⁴⁹, 59087, F₃, 22) (dual of [59087, 58938, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
        linear OA(3¹⁴¹, 59049, F₃, 22) (dual of [59049, 58908, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(3¹¹¹, 59049, F₃, 17) (dual of [59049, 58938, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(151−22, 151, large)-Net in Base 3 — Upper bound on s

There is no (129, 151, large)-net in base 3, because

20 times m-reduction [i] would yield (129, 131, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (151−22, 151, s)-Nets in Base 3

(151−22, 151, 5371)-Net over F3 — Constructive and digital

(151−22, 151, 21851)-Net over F3 — Digital

(151−22, 151, large)-Net in Base 3 — Upper bound on s

(151−22, 151, 5371)-Net over F₃ — Constructive and digital

(151−22, 151, 21851)-Net over F₃ — Digital