On theory and fast algorithms for error correction in residue number system product codes

Redundant Residue Number System Based Multicarrier DS-CDMA for Dynamic Multiple-Access in Cognitive Radios

“…Based onr r r (k) , the estimate to the message X k transmitted by the kth CRU can be recovered by the IRNST operation, for example, with the aid of the CRT [15]. Furthermore, when the RRNS is considered, the ratio statistic test (RST) assisted erasure-only decoding, error-correction-only decoding or the RST-assisted error-and-erasure decoding [16,17] can be employed to decoder r r (k) and recover the message X k transmitted.…”

“…In this RNS, any integer 0 ≤ X < M ≤ Q s i=1 mi can be uniquely and unambiguously represented by a residue sequence (r1, r2, ..., rs), where ri = X mod mi represents the residue of X with respect to mi, 0 ≤ ri < mi. Reversely, in this RNS, given an s-tuple residue sequence (r1, r2, ..., rs) with 0 ≤ ri < mi, i = 1, ..., s, according to the Chinese reminder theorem (CRT) [15], there exists one and only one integer 0 ≤ X < M satisfying ri = X mod mi. In other words, if (r1, r2, ..., rs) conveys a message X, it can be recovered uniquely and unambiguously with the aid of, for example, the Chinese reminder theorem.…”

“…For the sake of incorporating error control or making a RNSbased system fault-tolerant, redundant moduli can be appended to the information moduli, forming the so-called RRNS [8,11,[15][16][17]. Specifically, (q − s) moduli ms+1, ms+2, ..., mq, referred to as redundant moduli, can be added to the previously introduced RNS, in order to form a RRNS with q positive, pairwise relative prime integers.…”

Residue number system assisted fast frequency-hopped synchronous ultra-wideband spread-spectrum multiple-access: a design alternative to impulse radio

“…By contrast, the notation will be used in the context of MUD to be introduced in Section III, which will be referred to as the th user's MUD address. According to the Chinese remainder theorem (CRT) [27], for any given -tuple , where , there exists one and only one integer such that and , which allows us to uniquely recover the integer value of from the received residue digits. Again, represents the FH address of user .…”

Residue number system arithmetic assisted M-ary modulation

“…T HE SO-CALLED residue number system (RNS) [1]- [3] has two inherent features that render the RNS attractive in comparison to conventional weighted number systems, such as for example the binary representation. These two features are [2]: 1) the carry-free arithmetic and 2) the lack of ordered significance amongst the residue digits.…”