211 research outputs found

    On additive MDS codes over small fields

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    Let C C be a (n,q2k,n−k+1)q2 (n,q^{2k},n-k+1)_{q^2} additive MDS code which is linear over Fq {\mathbb F}_q . We prove that if n≥q+k n \geq q+k and k+1 k+1 of the projections of C C are linear over Fq2 {\mathbb F}_{q^2} then C C is linear over Fq2 {\mathbb F}_{q^2} . We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over Fq {\mathbb F}_q for q∈{4,8,9} q \in \{4,8,9\} . We also classify the longest additive MDS codes over F16 {\mathbb F}_{16} which are linear over F4 {\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∈{2,3} q \in \{ 2,3\} .Peer ReviewedPostprint (published version

    The equivalence of linear codes implies semi-linear equivalence

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    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

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    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

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    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

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    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 4×164\times 16 LMIMO-MBM is capable of transmitting 3232 bits of information per (complex) channel-use, with SER ≃10−5 \simeq 10^{-5} at Eb/N0≃−3.5E_b/N_0\simeq -3.5dB (or SER ≃10−4 \simeq 10^{-4} at Eb/N0=−4.5E_b/N_0=-4.5dB). 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
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