MinT - Best Known (128, 128+13, s)-Nets in Base 8

Best Known (128, 128+13, s)-Nets in Base 8

(128, 128+13, 5592400)-Net over F₈ — Constructive and digital

Digital (128, 141, 5592400)-net over F₈, using

8¹ times duplication [i] based on digital (127, 140, 5592400)-net over F₈, using
- trace code for nets [i] based on digital (57, 70, 2796200)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷⁰, 2796200, F₆₄, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(64⁷⁰, 8388601, F₆₄, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(64⁷⁰, 8388602, F₆₄, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(64²¹, 4194301, F₆₄, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²¹, 8388602, F₆₄, 6) (dual of [8388602, 8388581, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(128, 128+13, large)-Net over F₈ — Digital

Digital (128, 141, large)-net over F₈, using

t-expansion [i] based on digital (125, 141, large)-net over F₈, using
- 5 times m-reduction [i] based on digital (125, 146, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, large, F₈, 21) (dual of [large, large−146, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(128, 128+13, large)-Net in Base 8 — Upper bound on s

There is no (128, 141, large)-net in base 8, because

11 times m-reduction [i] would yield (128, 130, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (128, 128+13, s)-Nets in Base 8

(128, 128+13, 5592400)-Net over F8 — Constructive and digital

(128, 128+13, large)-Net over F8 — Digital

(128, 128+13, large)-Net in Base 8 — Upper bound on s

(128, 128+13, 5592400)-Net over F₈ — Constructive and digital

(128, 128+13, large)-Net over F₈ — Digital