Data link layer: framing, error control and MAC

Computer Science and Engineering (B.E./B.Tech) · Computer networks · free study notes for exams, placements and interviews

In short

The data link layer (OSI layer 2) turns a raw, error-prone bit pipe into a link that delivers *frames* between two directly connected nodes. Its four jobs are framing (marking where a frame starts and ends), error control (detecting or correcting bit errors and retransmitting), flow control (not drowning a slow receiver) and medium access control (deciding who transmits when a link is shared). It is the heaviest-scoring area of computer networks in GATE CS - sliding-window efficiency, CRC division, Hamming codes, minimum Ethernet frame size and ALOHA throughput appear almost every year - and it is standard interview ground for network engineer, NOC, embedded firmware and switching/routing roles at Cisco, Juniper, Nokia, Ericsson, Tejas Networks, C-DOT, HFCL, Jio, Airtel, Qualcomm and the service companies (TCS, Infosys, Wipro, HCLTech). On the job it explains CRC errors on an interface, runt and giant frames, duplex mismatch and MAC-table behaviour on a switch.

Explained

Services, sublayers and addressing

Position: sits above the physical layer, below the network layer. Scope is hop-to-hop (node to adjacent node), not end-to-end.
Three service classes: unacknowledged connectionless (Ethernet, most LANs - recovery left to TCP), acknowledged connectionless (802.11), and acknowledged connection-oriented (HDLC, older WAN links) with connection setup, numbered delivery and release.
Two sublayers (IEEE 802 model): LLC (802.2) - framing boundaries, flow and error control, multiplexing upper protocols via DSAP/SSAP; MAC - addressing and channel access. Modern Ethernet II uses a 2-byte EtherType instead of LLC headers (0x0800 IPv4, 0x0806 ARP, 0x86DD IPv6, 0x8100 VLAN tag).
MAC address: 48 bits, written as six hex bytes. Top 24 bits = OUI assigned by IEEE to the manufacturer; lower 24 bits = device serial. Bit 0 of the first byte is the I/G bit (0 = unicast, 1 = multicast; all-ones FF:FF:FF:FF:FF:FF = broadcast); bit 1 is the U/L bit (0 = globally unique, 1 = locally administered, as in Wi-Fi MAC randomisation).
Flow control: feedback-based (stop-and-wait, sliding window) at layer 2; rate-based control belongs to higher layers.

Framing

The physical layer delivers an undelimited bit stream; framing adds boundaries so a lost or corrupted frame damages only itself.
Character/byte count: a header field gives the frame length. Simple, but one corrupted count destroys synchronisation for every following frame - obsolete on its own.
Byte (character) stuffing: frame delimited by a flag byte; any flag or escape byte occurring inside the payload is preceded by an escape byte. PPP uses flag 0x7E and escape 0x7D, and XORs the stuffed byte with 0x20. Byte-oriented, so it assumes 8-bit characters.
Bit stuffing: used by HDLC. Flag = 01111110 (0x7E). The transmitter inserts a 0 after every five consecutive 1s in the data; the receiver removes a 0 that follows five 1s. Makes the frame length arbitrary (not a multiple of 8) and keeps the flag unique.
Physical-layer coding violations: reserved illegal line codes mark boundaries - Ethernet's preamble/SFD and 4B/5B or 8B/10B control symbols. Costs no data bits.
HDLC frame: Flag | Address | Control | Information | FCS (CRC-16/32) | Flag, with I-frames (information, carries N(S), N(R), P/F), S-frames (supervisory: RR, RNR, REJ, SREJ) and U-frames (unnumbered: control/management).
Ethernet (802.3) frame: Preamble 7 B of 10101010 | SFD 1 B 10101011 | Destination MAC 6 B | Source MAC 6 B | Type/Length 2 B | Payload 46-1500 B | FCS 4 B (CRC-32). Minimum frame 64 B and maximum 1518 B excluding preamble; interframe gap 96 bit times. A value ≤ 1500 in the third field means length (802.3), ≥ 1536 (0x0600) means EtherType. Padding fills short payloads to reach 46 B; a VLAN tag (802.1Q, 4 B) pushes the maximum to 1522 B.
Runt = frame below 64 B; giant/jumbo = above the MTU; both show in interface counters.

Error detection and correction

Error types: single-bit error (one bit flipped - rare on serial links) and burst error (a contiguous span in which the first and last bits are wrong - common, caused by noise lasting longer than one bit time). Higher bit rates mean more bits are hit by the same noise burst.
Redundancy: k data bits are sent as an n-bit codeword with r = n − k check bits. Hamming distance d(a,b) = number of differing bit positions; the minimum distance d_min of a code decides its power.
- Detect up to s errors: d_min ≥ s + 1. Correct up to t errors: d_min ≥ 2t + 1.
Simple parity (VRC): one bit makes the count of 1s even (or odd). Detects all odd numbers of errors, misses every even number. d_min = 2.
Two-dimensional parity (LRC + VRC): parity per row and per column; detects most bursts and corrects any single-bit error (row and column intersection), but misses four errors forming a rectangle.
Checksum: data split into m-bit words, added in 1's complement arithmetic, and the complement of the sum sent. Receiver adds everything including the checksum - an all-1s result means no detected error. Weaker than CRC (insensitive to word reordering); used by the IP header and by UDP/TCP (header plus data), not by Ethernet.
CRC (cyclic redundancy check): treat the k data bits as a polynomial, multiply by x^r (append r zeros), divide by the generator polynomial G(x) of degree r using modulo-2 (XOR) arithmetic, and append the r-bit remainder as the FCS. The receiver divides the whole received frame by G(x): remainder 0 means accept.
- Properties: detects all single-bit errors if G(x) has at least two terms; all burst errors of length ≤ r; all odd numbers of errors if (x + 1) is a factor of G(x); bursts of length r + 1 escape with probability 2^−(r−1).
- Standard generators: CRC-8 (ATM header), CRC-12, CRC-16 (x¹⁶+x¹⁵+x²+1), CRC-CCITT/CRC-16-ITU (x¹⁶+x¹²+x⁵+1), CRC-32 (x³²+…+x+1, generator 0x04C11DB7) used by Ethernet, 802.11, PPP, SATA and ZIP. Implemented in hardware by a shift register with XOR taps - which is why it is cheap at 100 Gbps.
Hamming code (forward error correction): r parity bits placed at positions 1, 2, 4, 8, … ; parity bit at position 2ⁱ checks every position whose binary representation has bit i set. Requires 2^r ≥ m + r + 1. The receiver recomputes the checks to form a syndrome; a non-zero syndrome is the binary position of the bad bit. Corrects one bit (d_min = 3); SECDED adds an overall parity bit for single-error-correct, double-error-detect (used in ECC RAM).
ARQ versus FEC: ARQ (retransmit on error) is efficient where a return path is cheap and delay is low - LANs, wired WANs. FEC (correct in place) is used where retransmission is impossible or costly - satellite, deep space, DVB, LTE/5G physical layer (convolutional, turbo, LDPC, Reed-Solomon codes).

Flow and error control protocols (ARQ)

Simplex/unrestricted protocol: assumes an error-free channel and an infinitely fast receiver - a teaching baseline only.
Stop-and-Wait: send one frame, wait for its ACK, then send the next. Needs 1-bit sequence numbers (0, 1) to detect a duplicate caused by a lost ACK. Window size 1 - dreadful on long, fast links.
Stop-and-Wait ARQ: adds a timer; on timeout the frame is resent. Lost ACK → duplicate frame, discarded using the sequence number.
Go-Back-N (GBN): sender window N, receiver window 1. The receiver discards any out-of-order frame; on timeout (or a REJ/NAK) the sender retransmits the lost frame and all frames after it. Sequence-number space: N ≤ 2ⁿ − 1 for n bits. Needs only one timer in practice, buffers N frames at the sender, none at the receiver. Good on low-error links.
Selective Repeat (SR): sender and receiver windows both N; out-of-order frames are buffered and only the missing frame is retransmitted (SREJ/NAK). Requires N ≤ 2ⁿ⁻¹ to prevent window overlap, per-frame timers, receiver buffering and resequencing. Best when the error rate is high or the bandwidth-delay product is large.
Piggybacking: carrying the acknowledgement for the reverse direction in the header of an outgoing data frame (N(R) field), saving a separate ACK frame; needs a bounded ACK-waiting timer.
Pipelining keeps the link busy while an ACK is in flight: utilisation rises from 1/(1+2a) to N/(1+2a), capped at 1.

Medium access control (MAC)

Needed only on a shared (multiple-access) medium; a full-duplex point-to-point link needs no MAC arbitration.
A. Random access (contention)
- Pure ALOHA: transmit whenever ready; collide, wait a random back-off, retry. Vulnerable period = 2 frame times; maximum throughput 0.184 (18.4%) at offered load G = 0.5.
- Slotted ALOHA: transmissions only at slot boundaries; vulnerable period = 1 frame time; maximum throughput 0.368 (1/e) at G = 1.
- CSMA: sense the carrier before transmitting. 1-persistent - transmit at once when idle (greedy, collides when two stations wait together); non-persistent - if busy, wait a random time and sense again (fewer collisions, more delay); p-persistent - in a slotted channel, transmit with probability p when idle. Collisions remain because of propagation delay.
- CSMA/CD (802.3, half-duplex Ethernet): sense, transmit, and keep *listening*; on collision send a 32-bit jam signal, abort, and back off by binary exponential backoff - wait k × slot time where k is random in [0, 2ⁿ − 1] for the n-th attempt, n capped at 10, and give up after 16 attempts. Requires frame transmission time ≥ 2 × propagation delay (one round trip), hence a minimum frame size.
- CSMA/CA (802.11): collisions cannot be detected on radio, so they are avoided - carrier sense plus DIFS wait, a random backoff counter that freezes while the channel is busy, mandatory ACK, and optional RTS/CTS handshake with NAV to handle the hidden-terminal problem.
B. Controlled access: reservation (slots claimed in a reservation frame), polling (a primary invites each secondary in turn - Bluetooth, older multidrop links; the primary is a single point of failure), token passing (a token circulates in a logical ring - Token Ring 802.5, FDDI, and token-bus style determinism in industrial networks).
C. Channelisation: FDMA (separate frequency bands), TDMA (separate time slots, needs synchronisation), CDMA (same band and time, orthogonal chip codes) - the basis of cellular access.
Switched Ethernet today: a hub repeats bits, so all ports form one collision domain; a switch learns source MACs into a CAM table, forwards per destination, floods unknown unicast and broadcast, and gives each port its own collision domain. With full duplex there are no collisions and CSMA/CD is disabled; 802.3x PAUSE frames provide link-level flow control. Switching modes: store-and-forward (checks FCS), cut-through, fragment-free.
Related layer-2 standards: 802.1D/802.1w STP/RSTP (breaks loops, since there is no TTL at layer 2), 802.1Q VLANs and trunking, 802.3ad/802.1AX link aggregation, 802.1X port-based authentication, 802.3af/at/bt Power over Ethernet.
Indian practice: IEEE 802.3 and 802.11 are adopted as-is; telecom equipment sold in India must clear MTCTE conformance testing administered by TEC under the DoT, with BIS marking for safety, and ITU-T recommendations apply on carrier links. Ethernet over fibre is the access technology for BharatNet and for most campus/ISP last miles; exact MTCTE product lists and schedules are revised periodically, so check the current TEC notification.

Key points

Data link layer - hop-to-hop delivery of frames; framing, error control, flow control, MAC, physical addressing.
Sublayers - LLC (802.2) handles link control and multiplexing; MAC handles addressing and channel access.
Bit stuffing - insert a 0 after five consecutive 1s (HDLC, flag 01111110); byte stuffing - escape the flag byte (PPP, 0x7E flag, 0x7D escape).
CRC - modulo-2 polynomial division; r check bits detect every burst up to length r; Ethernet FCS is CRC-32.
Hamming code - needs 2^r ≥ m + r + 1; parity bits at powers of 2; syndrome gives the erroneous bit position.
Hamming distance - detect s errors needs d_min ≥ s+1; correct t errors needs d_min ≥ 2t+1.
Parity - single parity misses all even numbers of errors; 2-D parity corrects one bit.
Stop-and-Wait - window 1, 1-bit sequence number, efficiency 1/(1+2a).
Go-Back-N - sender window N ≤ 2ⁿ − 1, receiver window 1, retransmits the whole window.
Selective Repeat - windows N ≤ 2ⁿ⁻¹ at both ends, retransmits only the lost frame, needs receiver buffering.
Piggybacking - ACK carried inside a reverse-direction data frame.
Pure ALOHA 18.4%, Slotted ALOHA 36.8% - maximum throughput, at G = 0.5 and G = 1 respectively.
CSMA/CD - sense, transmit, detect collision, jam 32 bits, binary exponential backoff, 16 attempts maximum.
Minimum frame size - 2 × propagation delay × bandwidth; 64 bytes (512 bits) for 10/100 Mbps Ethernet, so the slot time is 51.2 µs at 10 Mbps and 5.12 µs at 100 Mbps; 1 Gbps half duplex uses carrier extension to 512 bytes.
Ethernet frame - 6+6+2 header, 46-1500 payload, 4-byte FCS; 64 to 1518 bytes; 96-bit interframe gap.
MAC address - 48 bits, 24-bit OUI + 24-bit device ID; I/G bit marks multicast, U/L bit marks locally administered.
Hub versus switch - one collision domain versus one per port; full duplex removes CSMA/CD entirely.
STP - mandatory at layer 2 because frames carry no hop count, so a loop would broadcast-storm forever.

Formulas and examples

Transmission time Tt = L / B (L = frame length in bits, B = bandwidth in bits/s, Tt in s)
Propagation delay Tp = d / v (d = distance in m, v = signal speed ≈ 2 × 10⁸ m/s in copper/fibre ⇒ 5 µs per km)
a = Tp / Tt (dimensionless; bandwidth-delay product in frame units)
Stop-and-Wait efficiency η = 1 / (1 + 2a); with frame-error probability p, η = (1 − p) / (1 + 2a)
Sliding window efficiency η = N / (1 + 2a), capped at 1 (N = window size in frames)
Throughput = η × B; window for full utilisation N ≥ 1 + 2a
Sequence-number bits: GBN N ≤ 2ⁿ − 1; Selective Repeat N ≤ 2ⁿ⁻¹
Hamming code: 2^r ≥ m + r + 1 (m = data bits, r = parity bits, n = m + r)
CRC: transmitted = (data × x^r) with the remainder of division by G(x) appended; deg G(x) = r
Minimum frame length L_min = 2 × Tp × B = slot time × B
Pure ALOHA S = G e^−2G, max 0.184; Slotted ALOHA S = G e^−G, max 0.368 (G = attempts per frame time)
Backoff wait = k × slot time, k random in [0, 2ⁿ − 1] for the n-th collision (n ≤ 10)

Example 1 - Stop-and-Wait on a long link, and the window it needs. Link: B = 1 Mbps, frame L = 1000 bits, distance 4000 km, v = 2 × 10⁸ m/s.

Tt = 1000 / 10⁶ = 1 ms. Tp = 4 × 10⁶ / (2 × 10⁸) = 0.02 s = 20 ms.
a = 20 / 1 = 20, so 1 + 2a = 41.
Stop-and-Wait η = 1/41 = 2.44%; throughput = 0.0244 × 1 Mbps ≈ 24.4 kbps.
For 100% utilisation N ≥ 41 frames. GBN: 2ⁿ − 1 ≥ 41 ⇒ 2ⁿ ≥ 42 ⇒ n = 6 bits (N = 63 available). SR: 2ⁿ⁻¹ ≥ 41 ⇒ n = 7 bits.
With N = 4 only, η = 4/41 = 9.76% (throughput ≈ 97.6 kbps).

Example 2 - CRC generation and checking. Data 1101011011, generator G(x) = x⁴ + x + 1 → 10011, so r = 4.

Append 4 zeros: 11010110110000. Divide by 10011 using XOR:

11010110110000 ÷ 10011
11010 ^ 10011 = 01001  -> bring 1 -> 10011 ^ 10011 = 00000
bring 1 -> 00001 ; bring 0 -> 00010 ; bring 1 -> 00101 ; bring 1 -> 01011
bring 0 -> 10110 ^ 10011 = 00101 ; bring 0 -> 01010
bring 0 -> 10100 ^ 10011 = 00111 ; bring 0 -> 01110   (no bits left)

Remainder = 1110. Transmitted frame = 11010110111110.
At the receiver, dividing 11010110111110 by 10011 leaves remainder 0000 → accept. Any burst of 4 bits or fewer is guaranteed to be caught.

Example 3 - Hamming code: build it, then correct an error. Data m = 7 bits: 1011001. Check bits: 2^r ≥ 7 + r + 1 → r = 4 (16 ≥ 12), so n = 11.

Data at positions 3,5,6,7,9,10,11 = 1,0,1,1,0,0,1; parity at 1,2,4,8 (even parity).
P1 covers 1,3,5,7,9,11 → data 1,0,1,0,1 (three 1s) ⇒ P1 = 1.
P2 covers 2,3,6,7,10,11 → data 1,1,1,0,1 (four 1s) ⇒ P2 = 0.
P4 covers 4,5,6,7 → data 0,1,1 (two 1s) ⇒ P4 = 0.
P8 covers 8,9,10,11 → data 0,0,1 (one 1) ⇒ P8 = 1.
Codeword (positions 1→11) = 1 0 1 0 0 1 1 1 0 0 1.
Now flip position 6 in transit: received 1 0 1 0 0 0 1 1 0 0 1.
- C1 (1,3,5,7,9,11) = 1,1,0,1,0,1 → four 1s, even ⇒ 0.
- C2 (2,3,6,7,10,11) = 0,1,0,1,0,1 → three 1s, odd ⇒ 1.
- C4 (4,5,6,7) = 0,0,0,1 → odd ⇒ 1.
- C8 (8,9,10,11) = 1,0,0,1 → even ⇒ 0.
- Syndrome C8C4C2C1 = 0110 = 6 ⇒ flip bit 6 to recover the codeword.

Example 4 - Minimum Ethernet frame size. Classic 10BASE5 segment chain: worst-case one-way delay including repeaters = 25.6 µs, so round trip 2Tp = 51.2 µs at B = 10 Mbps.

L_min = 51.2 × 10⁻⁶ × 10 × 10⁶ = 512 bits = 64 bytes.
At 100 Mbps the 64-byte minimum is kept, so the slot time falls to 5.12 µs and the maximum half-duplex span shrinks to about 200 m of cable budget. At 1 Gbps half duplex the slot time is held by carrier extension padding to 512 bytes. In practice all modern Ethernet is full duplex and switched, so the limit is only historical - but it is why anything under 64 bytes is logged as a runt.
Reverse question: a 10 Mbps CSMA/CD LAN with 64-byte minimum frames allows 2Tp ≤ 51.2 µs ⇒ Tp ≤ 25.6 µs ⇒ at 5 µs/km, d ≤ 5.12 km of signal path.

Example 5 - ALOHA throughput with real numbers. Channel 1 Mbps, fixed frames of 1000 bits ⇒ frame time 1 ms. 50 stations each generate 20 frames/s.

Total offered load = 50 × 20 = 1000 frames/s ⇒ G = 1000 × 0.001 = 1.0.
Pure ALOHA: S = 1 × e⁻² = 0.1353 ⇒ ≈ 135 successful frames/s (135 kbps).
Slotted ALOHA: S = 1 × e⁻¹ = 0.3679 ⇒ ≈ 368 frames/s (368 kbps).
Best possible: pure ALOHA peaks at S = 0.184 (184 kbps) when G = 0.5; slotted at 0.368 (368 kbps) when G = 1. Pushing G beyond the peak reduces throughput - the collapse that motivated CSMA.

Protocol	Sender window	Receiver window	Retransmits	Seq-number bound	Best for
Stop-and-Wait ARQ	1	1	the one frame	1 bit	short, slow links
Go-Back-N ARQ	N	1	lost frame + all after	N ≤ 2ⁿ − 1	low error rate
Selective Repeat	N	N	only the lost frame	N ≤ 2ⁿ⁻¹	high error / high BDP

MAC scheme	Type	Max throughput	Determinism	Example
Pure ALOHA	random	18.4%	none	early satellite, RFID-like bursts
Slotted ALOHA	random	36.8%	none	reservation/request channels
CSMA/CD	random + sense	high at low load	none	legacy half-duplex Ethernet
CSMA/CA	random + avoid	~50-70% of PHY rate	none	Wi-Fi (802.11)
Token passing	controlled	near 100%	bounded delay	Token Ring, FDDI, industrial LANs
TDMA/FDMA/CDMA	channelised	fixed per user	strong	cellular, satellite trunks

Where it is used

Switching and router silicon: CRC-32 engines, MAC-address CAM/TCAM lookup, store-and-forward queues in products from Cisco, Juniper, Nokia, Arista, and Indian builds at Tejas Networks, HFCL, Sterlite Technologies and C-DOT (BharatNet and 4G/5G core gear).
ISP and enterprise network operations: Jio, Airtel, BSNL, Vodafone Idea and campus NOCs read interface counters - CRC/FCS errors point to a bad cable, SFP or EMI; late collisions and runts point to duplex mismatch or an over-long segment; STP reconvergence explains link flaps.
Embedded and automotive firmware: CAN bus (bit stuffing, CRC-15, arbitration by identifier priority), LIN, Modbus and industrial Ethernet in Bosch, Continental, Tata Elxsi, KPIT and L&T Technology Services projects; Ethernet MAC drivers on microcontrollers.
Wireless and telecom R&D: FEC and MAC scheduling work in 4G/5G at Qualcomm, Samsung R&D, Ericsson, Nokia and Reliance Jio Platforms; ARQ/HARQ design sits directly on these ideas.
Storage, memory and data integrity: CRC in SATA/PCIe/ZIP, Hamming SECDED in ECC server memory, Reed-Solomon in optical media and QR codes.
Exams and selection: GATE CS/IT, ISRO, BARC, DRDO, NIELIT, UGC-NET, PSU and bank-IO technical papers, plus first-round technical interviews for network/support engineer roles.

Interview questions

Q: Why does a frame need both a CRC at layer 2 and a checksum at layer 4? A: The CRC protects each hop's transmission over the physical medium, while the transport checksum is end-to-end and also guards against corruption inside routers, memory and buses that the per-hop CRC is recomputed over.
Q: What exactly is bit stuffing and why is it necessary? A: The transmitter inserts a 0 after five consecutive 1s so the data can never imitate the HDLC flag 01111110; the receiver strips it, keeping the frame delimiter unique while allowing arbitrary binary payloads.
Q: Why is the minimum Ethernet frame 64 bytes? A: A station must still be transmitting when the farthest station's collision signal returns, so the frame time must be at least one round-trip time - 51.2 µs at 10 Mbps, which is 512 bits.
Q: What errors can a CRC with a 16-bit FCS guarantee to detect? A: All single-bit errors, all odd numbers of errors if (x+1) divides the generator, all burst errors up to 16 bits, and longer bursts with probability of escape around 2^−15.
Q: Go-Back-N or Selective Repeat - which would you choose for a satellite link? A: Selective Repeat, because the bandwidth-delay product is huge and retransmitting a whole window for one lost frame wastes enormous capacity, at the cost of receiver buffering and per-frame timers.
Q: Why must the Selective Repeat window be at most half the sequence-number space? A: Otherwise, if all ACKs are lost, the retransmitted old frames occupy sequence numbers that fall inside the receiver's new window and would be accepted as fresh data.
Q: Does CSMA/CD still run on a modern switched network? A: No - full-duplex switched links have no shared medium and collisions are impossible, so CSMA/CD is disabled; it only matters on legacy half-duplex hub segments.
Q: Why does Wi-Fi use collision avoidance instead of collision detection? A: A radio cannot reliably listen while transmitting and the hidden-terminal problem means a sender may not hear the colliding station, so 802.11 avoids collisions with backoff, ACKs and optional RTS/CTS.
Q: What is piggybacking and what is its risk? A: Carrying an acknowledgement inside an outgoing data frame to save a frame; the risk is that if no reverse data appears, the ACK waits and may trigger a needless timeout, so an ACK timer bounds the delay.
Q: Slotted ALOHA doubles pure ALOHA's throughput - why? A: Synchronising transmissions to slot boundaries halves the vulnerable period from two frame times to one, so S = Ge^−G instead of Ge^−2G.
Q: Spanning Tree Protocol is layer 2 - why is it needed when IP has a TTL? A: Ethernet frames carry no hop count, so a physical loop would circulate broadcasts indefinitely and corrupt MAC tables; STP logically blocks redundant ports to produce a loop-free tree.
Q: An interface shows rising CRC errors but no collisions. What do you suspect? A: A physical-layer fault - damaged or out-of-spec cable, faulty SFP or port, excessive run length, or electrical interference - not contention.

Common mistakes

Using N instead of N/2 style slips in window formulas: writing GBN's bound as 2ⁿ⁻¹ and SR's as 2ⁿ − 1 (they are the other way round).
Forgetting that η = N/(1+2a) is capped at 1 and reporting efficiencies above 100%.
Mixing up a (= Tp/Tt) with its reciprocal, or forgetting the factor 2 in 1 + 2a (the ACK must travel back).
In CRC, appending r zeros but then transmitting the quotient, or sending the remainder with the zeros still attached - only the data followed by the r-bit remainder goes on the wire.
In Hamming codes, numbering positions from 0, or placing the data bits at positions 1, 2, 4, 8 - parity bits occupy the powers of two.
Treating the Ethernet 64-byte minimum and 1518-byte maximum as including the 8-byte preamble and SFD, or forgetting that padding is what makes a short payload legal.
Claiming CRC can correct errors (it only detects), or that single parity detects two-bit errors (it never does).
Confusing a collision domain with a broadcast domain: a switch splits collision domains, only a router or a VLAN boundary splits broadcast domains.