MinT - Best Known (156−25, 156, s)-Nets in Base 4

Best Known (156−25, 156, s)-Nets in Base 4

(156−25, 156, 5464)-Net over F₄ — Constructive and digital

Digital (131, 156, 5464)-net over F₄, using

4³ times duplication [i] based on digital (128, 153, 5464)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹⁵³, 5464, F₄, 25, 25) (dual of [(5464, 25), 136447, 26]-NRT-code), using
  - OOA 12-folding and stacking with additional row [i] based on linear OA(4¹⁵³, 65569, F₄, 25) (dual of [65569, 65416, 26]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹⁵³, 65570, F₄, 25) (dual of [65570, 65417, 26]-code), using
      - constructionÂ XX applied to C_e(24) âŠ‚ C_e(20) âŠ‚ C_e(18) [i] based on
        linear OA(4¹⁴⁵, 65536, F₄, 25) (dual of [65536, 65391, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(4¹²¹, 65536, F₄, 21) (dual of [65536, 65415, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(4¹¹³, 65536, F₄, 19) (dual of [65536, 65423, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(4⁵, 31, F₄, 3) (dual of [31, 26, 4]-code or 31-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
        a cap found by Tallinii [i]
        linear OA(4¹, 3, F₄, 1) (dual of [3, 2, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, 4, F₄, 1) (dual of [4, 3, 2]-code), using
        Reedâ€“Solomon code RS(3,4) [i]

(156−25, 156, 35849)-Net over F₄ — Digital

Digital (131, 156, 35849)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁶, 35849, F₄, 25) (dual of [35849, 35693, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁵⁶, 65580, F₄, 25) (dual of [65580, 65424, 26]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁵⁵, 65579, F₄, 25) (dual of [65579, 65424, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(4¹⁴⁵, 65537, F₄, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(4¹¹³, 65537, F₄, 19) (dual of [65537, 65424, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 4¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(4¹⁰, 42, F₄, 5) (dual of [42, 32, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(156−25, 156, large)-Net in Base 4 — Upper bound on s

There is no (131, 156, large)-net in base 4, because

23 times m-reduction [i] would yield (131, 133, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (156−25, 156, s)-Nets in Base 4

(156−25, 156, 5464)-Net over F4 — Constructive and digital

(156−25, 156, 35849)-Net over F4 — Digital

(156−25, 156, large)-Net in Base 4 — Upper bound on s

(156−25, 156, 5464)-Net over F₄ — Constructive and digital

(156−25, 156, 35849)-Net over F₄ — Digital