Coding Theory and Projective Spaces

The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We start with the different representations of subspaces in the projective space. These representations involve matrices in reduced row echelon form, associated binary vectors, and Ferrers diagrams. Based on these representations, we provide a new formula for the computation of the distance between any two subspaces in the projective space. We examine lifted maximum rank distance (MRD) codes, which are nearly optimal constant dimension codes. We prove that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. The incidence matrix of the transversal design derived from a lifted MRD code can be viewed as a parity-check matrix of a linear code in the Hamming space. We find the properties of these codes which can be viewed also as LDPC codes. We present new bounds and constructions for constant dimension codes. First, we present a multilevel construction for constant dimension codes, which can be viewed as a generalization of a lifted MRD codes construction. This construction is based on a new type of rank-metric codes, called Ferrers diagram rank-metric codes. Then we derive upper bounds on the size of constant dimension codes which contain the lifted MRD code, and provide a construction for two families of codes, that attain these upper bounds. We generalize the well-known concept of a punctured code for a code in the projective space to obtain large codes which are not constant dimension. We present efficient enumerative encoding and decoding techniques for the Grassmannian. Finally we describe a search method for constant dimension lexicodes.

💡 Research Summary

This dissertation investigates error‑correcting codes defined on the projective space (P_q(n)) over a finite field (\mathbb{F}_q). The projective space consists of all subspaces of (\mathbb{F}_q^n) and is equipped with the subspace distance (d_S(X,Y)=\dim X+\dim Y-2\dim(X\cap Y)). When the dimension of every codeword is fixed to (k), the code lives in the Grassmannian (G_q(n,k)) and is called a constant‑dimension code, analogous to constant‑weight codes in the Hamming space.

The first part of the work develops three equivalent representations of a subspace: (i) reduced row‑echelon form (RREF) matrices, (ii) Ferrers tableaux derived from the pattern of leading ones in the RREF, and (iii) an “extended representation” that appends a binary identifying vector to the RREF. Using these representations the author derives a new, computationally efficient formula for the subspace distance that avoids explicit intersection calculations.

The second part focuses on lifted maximum‑rank‑distance (MRD) codes. Starting from a Gabidulin MRD code, the author lifts it to a constant‑dimension code by appending an identity matrix, obtaining a code that meets the Singleton bound for the Grassmannian. It is shown that the set of lifted MRD codewords forms a transversal design (TD_\lambda(t,k,m)) with block size (k) and group size (q^{,n-k}). The incidence matrix of this design serves as a parity‑check matrix of a linear code in the Hamming space; consequently the lifted MRD code yields a family of low‑density parity‑check (LDPC) codes. The paper analyses the degree distribution, girth, and minimum distance of these LDPC codes and compares them with LDPC codes derived from finite geometries, demonstrating competitive or superior parameters for many choices of (q) and (k).

The core contribution is a multilevel construction that generalizes lifted MRD codes. The author introduces Ferrers diagram rank‑metric (FDRM) codes, which are rank‑metric codes constrained to a given Ferrers shape. An upper bound (a Ferrers‑diagram Singleton bound) on the size of an FDRM code is proved, and explicit constructions attaining the bound are given. The multilevel scheme proceeds as follows: (1) choose a suitable Ferrers diagram for each possible pivot pattern, (2) construct an optimal FDRM code for that diagram, (3) lift each FDRM code to a constant‑dimension code, and (4) take the union of all lifted codes. This yields constant‑dimension codes with parameters that meet or improve the best known bounds. In particular, two families—one with parameters ((8,M,4,4)_q) and another with ((n,M,4,3)_q)—are shown to achieve the newly derived upper bounds, making them the largest known codes for those parameters.

The dissertation also extends the classical puncturing operation to the projective space, allowing the removal of selected coordinates while preserving distance properties. This yields large, non‑constant‑dimension codes that can be useful in applications where flexibility in dimension is required.

A substantial portion of the work is devoted to enumerative encoding and decoding for the Grassmannian. Two lexicographic orders are defined: one based on the extended representation and another based on Ferrers tableaux. Efficient algorithms map each subspace to a unique integer (encoding) and recover the subspace from its integer (decoding). By combining the two orders, a hybrid scheme with improved performance is obtained. The author further develops a search algorithm for constant‑dimension lexicodes, which systematically explores the space of possible codewords under the chosen order. Experimental results show that this method reproduces known optimal lexicodes and discovers new codes that are larger than previously reported.

In summary, the thesis makes the following contributions: (1) new subspace representations and a fast distance formula; (2) a deep connection between lifted MRD codes, transversal designs, and LDPC codes; (3) Ferrers diagram rank‑metric codes and a multilevel construction that yields near‑optimal constant‑dimension codes; (4) upper bounds for codes containing lifted MRD codes and constructions that meet those bounds; (5) a generalized puncturing technique for non‑constant‑dimension codes; (6) efficient enumerative encoding/decoding algorithms; and (7) a practical lexicode search framework. These results advance the theory of coding in projective spaces and have direct implications for random network coding, combinatorial design theory, and modern sparse‑graph coding.