Stefano Della Fiore

Stefano Della Fiore

We derive refined entropy upper bounds for $q$-ary $B_2$ codes by exploiting the Fourier structure of the i.i.d. difference distribution $D=X-Y$. Since the pmf of $D$ is an autocorrelation, its Fourier series is a nonnegative trigonometric polynomial of degree at most $q-1$. This leads to a natural convex relaxation over candidate difference distributions, equivalently expressible through an infinite family of positive semidefinite Toeplitz constraints. The resulting formulation admits a simple Gram interpretation and yields certified upper bounds through truncated semidefinite programs. Combined with the prefix-suffix method, this gives improved asymptotic rate upper bounds for $q$-ary $B_2$ codes; in particular, for $q\in\{9,10,11,12,13\}$ the resulting values improve on the best bounds known in the literature. We also study binary constant-weight $B_2$ codes. Extending the distance-distribution method of Cohen, Litsyn, and Zémor to the constant-weight setting, and combining it with Litsyn's asymptotic linear-programming bound for constant-weight codes, we derive a new upper bound on the constant-weight $B_2$ rate.

Stefano Della Fiore

We study a Pareto-type notion of finite-block optimality for injective source codes, where two codes are compared through the full sequence of expected block lengths. As a concrete and fully analyzable test case, we revisit the four-symbol constrained Markov source introduced by Dalai and Leonardi in their "meaningful example'' on constrained-source decodability. For each admissible nonempty string $u=x_1^m \in \mathscr{A} \subset \mathscr{X}^+$, let $$ K(u):=-\log_2 \mathbb{P}(X_1^m=u) $$ denote its information cost. We construct a canonical injective binary mapping $C:\mathscr{A} \to \{0,1\}^+$ by ordering admissible strings by increasing $K(u)$, then by length and lexicographic order, and assigning binary strings in shortlex order. For the length-$n$ block $X_1^n$ we prove $$ \mathbb{E}[|C(X_1)|]=\tfrac32, \qquad \mathbb{E}[|C(X_1^n)|]<\tfrac32\,n\quad (n\ge 2). $$ Moreover, for every fixed $$ 0<c<\frac{\sqrt2}{18\sqrtπ} $$ we have $$ \mathbb{E}[|C(X_1^n)|]\le \tfrac32\,n-\frac{c}{\sqrt n} $$ for all sufficiently large $n$. Thus, for this source, the reversible Dalai-Leonardi code is not Pareto-optimal with respect to finite-block average length. The proof is based on an exact enumeration of admissible strings by information cost and on a shortlex gap identity implying that each cost class splits evenly between lengths $K(u)-1$ and $K(u)$. The example is simple, but it already exhibits the kind of finite-block Pareto comparison that seems natural for injective source coding under source constraints.

Stefano Della Fiore, Alessandro Gnutti, Marco Dalai et al.

Traditional image compression methods aim to reconstruct images for human perception, prioritizing visual fidelity over task relevance. In contrast, Coding for Machines focuses on preserving information essential for automated understanding. Building on this principle, we present an end-to-end compression framework that retains text-specific features for Optical Character Recognition (OCR). The encoder operates at roughly half the computational cost of the OCR module, making it suitable for resource-limited devices. When on-device OCR is infeasible, images can be efficiently compressed and later decoded to recover textual content. Experiments show significant improvements in text extraction accuracy at low bitrates, even outperforming OCR on uncompressed images. We further extend this study to general-purpose encoders, exploring their capacity to preserve hidden semantics under extreme compression. Instead of optimizing for visual fidelity, we examine whether compact, visually degraded representations can retain recoverable meaning through learned enhancement and recognition modules. Results demonstrate that semantic information can persist despite severe compression, bridging text-oriented compression and general-purpose semantic preservation in machine-centered image coding.