Codevektors

Codevektors are a concept in coding theory describing augmented codewords that combine a traditional codeword with an accompanying feature vector. They are used to model systems where information is transmitted together with metadata or state information, allowing joint representation of data content and auxiliary attributes such as timing, channel state, or attribute tags.

Formally, let F be a finite field and C ⊆ F^n a code. A codevector is an element (c, v) with c ∈ C and v ∈ V, where V is a vector space over F that encodes side information. The set of codevectors forms a product space C × V. A joint distance can be defined, for example d((c1, v1), (c2, v2)) = d_C(c1, c2) + λ d_V(v1, v2), where d_C is a code distance and d_V measures difference in the feature vectors. Decoding can be performed with a joint decoder that outputs both an estimated codeword and the corresponding feature vector.

Variants and properties of codevektors depend on the choice of C and V. If C is linear and V is a linear space with an appropriate addition, the codevector set can inherit a linear structure in the product space. The concept generalizes ideas from concatenated codes and joint source-channel coding and is compatible with standard decoding techniques such as syndrome decoding when extended to the product space.

Applications and status: The notion of codevectors is primarily explored in theoretical contexts and in niche applications that require simultaneous transmission of data and metadata. Practical implementation considerations include increased decoding complexity and management of the feature-space dimensionality. See also coding theory, error-correcting codes, product codes, and joint source-channel coding.