Using the fact that every worldsheet may be ruled by two copies of worldlines, a restricted version of Weyl’s construction of representations of algebras is used to extend the recent classification of off-shell supermultiplets of N-extended world-line supersymmetry to usual off-shell and also unidextrous (on-the-half-shell) su-permultiplets of worldsheet (p, q)-supersymmetry with no central extension. In the process, a new class of error-correcting (even-split doubly-even linear block) codes is introduced and classified for p+q ≤ 8, providing a graphical method for classification of such codes, and thereby of so-projected supermultiplets.

There exist myriads of off-shell worldline supermultiplets for (N ≤ 32)-extended su-persymmetry in which every supercharge maps a component field to precisely one other component field or its derivative. A subset of these extends to off-shell world-sheet (p, q)-supersymmetry and is characterized by the twin theorems 2.1 and 2.2 in this note. The evasion of the obstruction defined in these theorems is conjectured to be sufficient for a worldline supermultiplet to extend to worldsheet supersymmetry; it is also a necessary filter for dimensional extension to higher-dimensional space-time. We show explicitly how to “re-engineer ” an Adinkra—if permitted by the twin theorems 2.1 and 2.2—so as to depict an off-shell supermultiplet of worldsheet (p, q)-supersymmetry.