MinT - Best Known (148−12, 148, s)-Nets in Base 8

Best Known (148−12, 148, s)-Nets in Base 8

(148−12, 148, 5608790)-Net over F₈ — Constructive and digital

Digital (136, 148, 5608790)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (13, 17, 16390)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁷, 16390, F₈, 4, 4) (dual of [(16390, 4), 65543, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁷, 16390, F₈, 3, 4) (dual of [(16390, 3), 49153, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹⁷, 32780, F₈, 4) (dual of [32780, 32763, 5]-code), using
        constructionÂ X4 applied to C_e(3) âŠ‚ C_e(1) [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(8⁶, 32768, F₈, 2) (dual of [32768, 32762, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(8¹¹, 12, F₈, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,8)), using
        dual of repetition code with length 12 [i]
        linear OA(8¹, 12, F₈, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, 511, F₈, 1) (dual of [511, 510, 2]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,0], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [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 (78, 90, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁰, 2796200, F₈, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁰, 8388601, F₈, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁰, 8388602, F₈, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁵, 4194301, F₆₄, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁵, 8388602, F₆₄, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(148−12, 148, 5657680)-Net in Base 8 — Constructive

(136, 148, 5657680)-net in base 8, using

base change [i] based on digital (99, 111, 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 (56, 68, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(148−12, 148, large)-Net over F₈ — Digital

Digital (136, 148, large)-net over F₈, using

t-expansion [i] based on digital (131, 148, large)-net over F₈, using
- 5 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(148−12, 148, large)-Net in Base 8 — Upper bound on s

There is no (136, 148, large)-net in base 8, because

10 times m-reduction [i] would yield (136, 138, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (148−12, 148, s)-Nets in Base 8

(148−12, 148, 5608790)-Net over F8 — Constructive and digital

(148−12, 148, 5657680)-Net in Base 8 — Constructive

(148−12, 148, large)-Net over F8 — Digital

(148−12, 148, large)-Net in Base 8 — Upper bound on s

(148−12, 148, 5608790)-Net over F₈ — Constructive and digital

(148−12, 148, large)-Net over F₈ — Digital