MinT - Best Known (125−9, 125, s)-Nets in Base 4

Best Known (125−9, 125, s)-Nets in Base 4

(125−9, 125, large)-Net over F₄ — Constructive and digital

Digital (116, 125, large)-net over F₄, using

4⁹ times duplication [i] based on digital (107, 116, large)-net over F₄, using
- trace code for nets [i] based on digital (20, 29, 2097406)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256²⁹, 2097406, F₂₅₆, 12, 9) (dual of [(2097406, 12), 25168843, 10]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(256²⁹, 2097407, F₂₅₆, 4, 9) (dual of [(2097407, 4), 8389599, 10]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(256⁴, 257, F₂₅₆, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;1024,256) [i]
        linear OOA(256²⁵, 2097150, F₂₅₆, 4, 9) (dual of [(2097150, 4), 8388575, 10]-NRT-code), using
        OOA 4-folding [i] based on linear OA(256²⁵, 8388600, F₂₅₆, 9) (dual of [8388600, 8388575, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁵, large, F₂₅₆, 9) (dual of [large, large−25, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(125−9, 125, large)-Net in Base 4 — Upper bound on s

There is no (116, 125, large)-net in base 4, because

7 times m-reduction [i] would yield (116, 118, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]