MinT - Best Known (157−13, 157, s)-Nets in Base 8

Best Known (157−13, 157, s)-Nets in Base 8

(157−13, 157, 5608795)-Net over F₈ — Constructive and digital

Digital (144, 157, 5608795)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (14, 18, 16395)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁸, 16395, F₈, 4, 4) (dual of [(16395, 4), 65562, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁸, 16395, F₈, 3, 4) (dual of [(16395, 3), 49167, 5]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8², 9, F₈, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;25,8) [i]
        linear OOA(8¹⁶, 16386, F₈, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(8¹⁶, 32772, F₈, 4) (dual of [32772, 32756, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹⁶, 32773, F₈, 4) (dual of [32773, 32757, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(8¹⁶, 32768, F₈, 4) (dual of [32768, 32752, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8¹¹, 32768, F₈, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (35, 41, 2796200)-net over F₈, using
  - s-reduction based on digital (35, 41, 2796201)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁴¹, 2796201, F₈, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(8⁴¹, large, F₈, 6) (dual of [large, large−41, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on 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]

(157−13, 157, 5657680)-Net in Base 8 — Constructive

(144, 157, 5657680)-net in base 8, using

8¹ times duplication [i] based on (143, 156, 5657680)-net in base 8, using
- base change [i] based on digital (104, 117, 5657680)-net over F₁₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (8, 12, 65280)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16¹², 65280, F₁₆, 4, 4) (dual of [(65280, 4), 261108, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(16¹², 130560, F₁₆, 4) (dual of [130560, 130548, 5]-code), using
        trace code [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
    2. digital (25, 31, 2796200)-net over F₁₆, using
      - s-reduction based on digital (25, 31, 2796201)-net over F₁₆, using
        net defined by OOA [i] based on linear OOA(16³¹, 2796201, F₁₆, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(16³¹, large, F₁₆, 6) (dual of [large, large−31, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. digital (61, 74, 2796200)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁷⁴, 2796200, F₁₆, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁷⁴, 8388601, F₁₆, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁷⁴, 8388602, F₁₆, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
        trace code [i] based on linear OOA(256³⁷, 4194301, F₂₅₆, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁷, 8388602, F₂₅₆, 13) (dual of [8388602, 8388565, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁷, large, F₂₅₆, 13) (dual of [large, large−37, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(157−13, 157, large)-Net over F₈ — Digital

Digital (144, 157, large)-net over F₈, using

11 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(157−13, 157, large)-Net in Base 8 — Upper bound on s

There is no (144, 157, large)-net in base 8, because

11 times m-reduction [i] would yield (144, 146, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (157−13, 157, s)-Nets in Base 8

(157−13, 157, 5608795)-Net over F8 — Constructive and digital

(157−13, 157, 5657680)-Net in Base 8 — Constructive

(157−13, 157, large)-Net over F8 — Digital

(157−13, 157, large)-Net in Base 8 — Upper bound on s

(157−13, 157, 5608795)-Net over F₈ — Constructive and digital

(157−13, 157, large)-Net over F₈ — Digital