On additive MDS codes over small fields

Let $C$ be a $(n,q^{2k},n-k+1)_{q^2}$ additive MDS code which is linear over ${\mathbb F}_q$. We prove that if $n \geq q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ is linear over ${\mathbb F}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb F}_q$ for $q \in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb F}_{16}$ which are linear over ${\mathbb F}_4$. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $q \in \{ 2,3\}$.Peer ReviewedPostprint (published version

The equivalence of linear codes implies semi-linear equivalence

The version of record is available online at: 10.1007/s10623-022-01055-xWe prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.Peer ReviewedPostprint (author's final draft

Wet paper codes and the dual distance in steganography

In 1998 Crandall introduced a method based on coding theory to secretly embed a message in a digital support such as an image. Later Fridrich et al. improved this method to minimize the distortion introduced by the embedding; a process called wet paper. However, as previously emphasized in the literature, this method can fail during the embedding step. Here we find sufficient and necessary conditions to guarantee a successful embedding by studying the dual distance of a linear code. Since these results are essentially of combinatorial nature, they can be generalized to systematic codes, a large family containing all linear codes. We also compute the exact number of solutions and point out the relationship between wet paper codes and orthogonal arrays

Additive MDS codes

We prove that an additive code over a finite field which has a few projections which are equivalent to a linear code is itself equivalent to a linear code, providing the code is not too short.Postprint (published version

Media-Based MIMO: A New Frontier in Wireless Communications

The idea of Media-based Modulation (MBM), is based on embedding information in the variations of the transmission media (channel state). This is in contrast to legacy wireless systems where data is embedded in a Radio Frequency (RF) source prior to the transmit antenna. MBM offers several advantages vs. legacy systems, including "additivity of information over multiple receive antennas", and "inherent diversity over a static fading channel". MBM is particularly suitable for transmitting high data rates using a single transmit and multiple receive antennas (Single Input-Multiple Output Media-Based Modulation, or SIMO-MBM). However, complexity issues limit the amount of data that can be embedded in the channel state using a single transmit unit. To address this shortcoming, the current article introduces the idea of Layered Multiple Input-Multiple Output Media-Based Modulation (LMIMO-MBM). Relying on a layered structure, LMIMO-MBM can significantly reduce both hardware and algorithmic complexities, as well as the training overhead, vs. SIMO-MBM. Simulation results show excellent performance in terms of Symbol Error Rate (SER) vs. Signal-to-Noise Ratio (SNR). For example, a $4\times 16$ LMIMO-MBM is capable of transmitting $32$ bits of information per (complex) channel-use, with SER $\simeq 10^{-5}$ at $E_b/N_0\simeq -3.5$dB (or SER $\simeq 10^{-4}$ at $E_b/N_0=-4.5$dB). This performance is achieved using a single transmission and without adding any redundancy for Forward-Error-Correction (FEC). This means, in addition to its excellent SER vs. energy/rate performance, MBM relaxes the need for complex FEC structures, and thereby minimizes the transmission delay. Overall, LMIMO-MBM provides a promising alternative to MIMO and Massive MIMO for the realization of 5G wireless networks.Comment: 26 pages, 11 figures, additional examples are given to further explain the idea of Media-Based Modulation. Capacity figure adde