The Ladder of Spectral Gaps F; Channel Capacity and Rate Gap– A Convergence Theorem for Constraint Network Information Geometry
changzheng zhou, ZIQING zhou
Shannon’s channel coding theorem establishes the upper bound of reliable communication rate as the channel capacity C, while the rate R of practical codingschemes is usually strictly lower than C, and the difference between them constitutes the rate gap. Within the framework of constraint network informationgeometry, this paper encodes a discrete memoryless channel and a linear blockcode as a channel constraint network and a coding constraint network respectively,and proposes to strictly define the information dimension by the volume growthrate of the rigid kernel under the Fisher information metric. The core conclusionshows that the Shannon gap C−R is asymptotically exactly equal to the logarithmnormalized limit of the ratio of Fisher volumes of the rigid kernels of the channelconstraint network and the coding constraint network, i.e., the difference of thevolume growth rates of the two networks. Under the local asymptotic normalitycondition, this gap satisfies L2 = 2(C −R)+O(n−1/2logn), where L is the squareof the Fisher geodesic distance. For the repetition code (3,1) over the binary symmetric channel, the analytical probability distributions of all eight output patternsare given, the finite-length deviation of the KL divergence from the asymptoticgap is computed, and its sources are analyzed. For the Hamming (7,4) code overthe binary erasure channel, by combinatorially enumerating all one hundred andtwenty-eight erasure patterns over finite fields, the average residual non-recoverabledimension and its deviation from the asymptotic prediction n(C −R) are preciselygiven, verifying the error bound O(1/√n). Finally, a roadmap is provided to extendthe volume compression framework to LDPC codes via the peeling algorithm andto Polar codes via the polarization transform, and the applicability boundaries ofthe current framework in the cases of asymmetric channels, continuous alphabets,and non-linear codes are honestly declared.
•May 29, 2026•0 citations