diff --git a/specs/altair/beacon-chain.md b/specs/altair/beacon-chain.md
index 7412a8490..6597751da 100644
--- a/specs/altair/beacon-chain.md
+++ b/specs/altair/beacon-chain.md
@@ -137,8 +137,8 @@ This patch updates a few configuration values to move penalty parameters closer
 
 | Name | Value | Description |
 | - | - | - |
-| `INACTIVITY_SCORE_BIAS` | `uint64(4)` | score points per inactive epoch |
-| `INACTIVITY_SCORE_RECOVERY_RATE` | `uint64(16)` | score points per recovering epoch |
+| `INACTIVITY_SCORE_BIAS` | `uint64(2**2)` (= 4) | score points per inactive epoch |
+| `INACTIVITY_SCORE_RECOVERY_RATE` | `uint64(2**4)` (= 16) | score points per leak-free epoch |
 
 ## Containers
 
@@ -157,8 +157,7 @@ class BeaconBlockBody(Container):
     attestations: List[Attestation, MAX_ATTESTATIONS]
     deposits: List[Deposit, MAX_DEPOSITS]
     voluntary_exits: List[SignedVoluntaryExit, MAX_VOLUNTARY_EXITS]
-    # [New in Altair]
-    sync_aggregate: SyncAggregate
+    sync_aggregate: SyncAggregate  # [New in Altair]
 ```
 
 #### `BeaconState`
@@ -266,10 +265,7 @@ def has_flag(flags: ParticipationFlags, flag_index: int) -> bool:
 ```python
 def get_next_sync_committee_indices(state: BeaconState) -> Sequence[ValidatorIndex]:
     """
-    Return the sequence of sync committee indices (which may include duplicate indices)
-    for the next sync committee, given a ``state`` at a sync committee period boundary.
-
-    Note: Committee can contain duplicate indices for small validator sets (< SYNC_COMMITTEE_SIZE + 128)
+    Return the sync committee indices, with possible duplicates, for the next sync committee.
     """
     epoch = Epoch(get_current_epoch(state) + 1)
 
@@ -292,21 +288,12 @@ def get_next_sync_committee_indices(state: BeaconState) -> Sequence[ValidatorInd
 
 #### `get_next_sync_committee`
 
+*Note*: The function `get_next_sync_committee` should only be called at sync committee period boundaries.
+
 ```python
 def get_next_sync_committee(state: BeaconState) -> SyncCommittee:
     """
-    Return the *next* sync committee for a given ``state``.
-
-    ``SyncCommittee`` contains an aggregate pubkey that enables
-    resource-constrained clients to save some computation when verifying
-    the sync committee's signature.
-
-    ``SyncCommittee`` can also contain duplicate pubkeys, when ``get_next_sync_committee_indices``
-    returns duplicate indices. Implementations must take care when handling
-    optimizations relating to aggregation and verification in the presence of duplicates.
-
-    Note: This function should only be called at sync committee period boundaries by ``process_sync_committee_updates``
-    as ``get_next_sync_committee_indices`` is not stable within a given period.
+    Return the next sync committee, with possible pubkey duplicates.
     """
     indices = get_next_sync_committee_indices(state)
     pubkeys = [state.validators[index].pubkey for index in indices]
@@ -325,14 +312,12 @@ def get_base_reward_per_increment(state: BeaconState) -> Gwei:
 
 *Note*: The function `get_base_reward` is modified with the removal of `BASE_REWARDS_PER_EPOCH` and the use of increment based accounting.
 
+*Note*: On average an optimally performing validator earns one base reward per epoch.
+
 ```python
 def get_base_reward(state: BeaconState, index: ValidatorIndex) -> Gwei:
     """
     Return the base reward for the validator defined by ``index`` with respect to the current ``state``.
-
-    Note: An optimally performing validator can earn one base reward per epoch over a long time horizon.
-    This takes into account both per-epoch (e.g. attestation) and intermittent duties (e.g. block proposal
-    and sync committees).
     """
     increments = state.validators[index].effective_balance // EFFECTIVE_BALANCE_INCREMENT
     return Gwei(increments * get_base_reward_per_increment(state))
@@ -559,6 +544,8 @@ def process_deposit(state: BeaconState, deposit: Deposit) -> None:
 
 #### Sync committee processing
 
+*Note*: The function `process_sync_committee` is new.
+
 ```python
 def process_sync_committee(state: BeaconState, aggregate: SyncAggregate) -> None:
     # Verify sync committee aggregate signature signing over the previous slot block root
@@ -627,17 +614,17 @@ def process_justification_and_finalization(state: BeaconState) -> None:
 
 ```python
 def process_inactivity_updates(state: BeaconState) -> None:
-    # Score updates based on previous epoch participation, skip genesis epoch
+    # Skip the genesis epoch as score updates are based on the previous epoch participation
     if get_current_epoch(state) == GENESIS_EPOCH:
         return
 
     for index in get_eligible_validator_indices(state):
-        # Increase inactivity score of inactive validators
+        # Increase the inactivity score of inactive validators
         if index in get_unslashed_participating_indices(state, TIMELY_TARGET_FLAG_INDEX, get_previous_epoch(state)):
             state.inactivity_scores[index] -= min(1, state.inactivity_scores[index])
         else:
             state.inactivity_scores[index] += INACTIVITY_SCORE_BIAS
-        # Decrease the score of all validators for forgiveness when not during a leak
+        # Decrease the inactivity score of all eligible validators during a leak-free epoch
         if not is_in_inactivity_leak(state):
             state.inactivity_scores[index] -= min(INACTIVITY_SCORE_RECOVERY_RATE, state.inactivity_scores[index])
 ```
diff --git a/specs/altair/validator.md b/specs/altair/validator.md
index 2258452c9..844c7021b 100644
--- a/specs/altair/validator.md
+++ b/specs/altair/validator.md
@@ -282,7 +282,12 @@ def get_sync_committee_message(state: BeaconState,
     signing_root = compute_signing_root(block_root, domain)
     signature = bls.Sign(privkey, signing_root)
 
-    return SyncCommitteeMessage(slot=state.slot, validator_index=validator_index, signature=signature)
+    return SyncCommitteeMessage(
+        slot=state.slot,
+        beacon_block_root=block_root,
+        validator_index=validator_index,
+        signature=signature,
+    )
 ```
 
 ##### Broadcast sync committee message
diff --git a/tests/core/pyspec/eth2spec/test/altair/transition/test_transition.py b/tests/core/pyspec/eth2spec/test/altair/transition/test_transition.py
index 20e13b9ba..62740df4e 100644
--- a/tests/core/pyspec/eth2spec/test/altair/transition/test_transition.py
+++ b/tests/core/pyspec/eth2spec/test/altair/transition/test_transition.py
@@ -1,7 +1,7 @@
 import random
 from eth2spec.test.context import fork_transition_test
 from eth2spec.test.helpers.constants import PHASE0, ALTAIR
-from eth2spec.test.helpers.state import state_transition_and_sign_block, next_slot, next_epoch_via_block
+from eth2spec.test.helpers.state import state_transition_and_sign_block, next_slot, next_epoch_via_signed_block
 from eth2spec.test.helpers.block import build_empty_block_for_next_slot, build_empty_block, sign_block
 from eth2spec.test.helpers.attestations import next_slots_with_attestations
 
@@ -261,12 +261,12 @@ def _run_transition_test_with_attestations(state,
     assert current_epoch == spec.GENESIS_EPOCH
 
     # skip genesis epoch to avoid dealing with some edge cases...
-    block = next_epoch_via_block(spec, state)
+    block = next_epoch_via_signed_block(spec, state)
 
     # regular state transition until fork:
     fill_cur_epoch = False
     fill_prev_epoch = True
-    blocks = [pre_tag(sign_block(spec, state, block))]
+    blocks = [pre_tag(block)]
     current_epoch = spec.get_current_epoch(state)
     for _ in range(current_epoch, fork_epoch - 1):
         _, blocks_in_epoch, state = next_slots_with_attestations(
@@ -414,8 +414,8 @@ def test_transition_with_no_attestations_until_after_fork(state, fork_epoch, spe
 
     # continue regular state transition but add attestations
     # for enough epochs to finalize the ``fork_epoch``
-    block = next_epoch_via_block(post_spec, state)
-    blocks.append(post_tag(sign_block(post_spec, state, block)))
+    block = next_epoch_via_signed_block(post_spec, state)
+    blocks.append(post_tag(block))
     for _ in range(4):
         _, blocks_in_epoch, state = next_slots_with_attestations(
             post_spec,
diff --git a/tests/core/pyspec/eth2spec/test/helpers/state.py b/tests/core/pyspec/eth2spec/test/helpers/state.py
index ef09c6e07..05f0e9013 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/state.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/state.py
@@ -58,11 +58,19 @@ def next_epoch(spec, state):
         spec.process_slots(state, slot)
 
 
-def next_epoch_via_block(spec, state):
+def next_epoch_via_block(spec, state, insert_state_root=False):
     """
     Transition to the start slot of the next epoch via a full block transition
     """
-    return apply_empty_block(spec, state, state.slot + spec.SLOTS_PER_EPOCH - state.slot % spec.SLOTS_PER_EPOCH)
+    block = apply_empty_block(spec, state, state.slot + spec.SLOTS_PER_EPOCH - state.slot % spec.SLOTS_PER_EPOCH)
+    if insert_state_root:
+        block.state_root = state.hash_tree_root()
+    return block
+
+
+def next_epoch_via_signed_block(spec, state):
+    block = next_epoch_via_block(spec, state, insert_state_root=True)
+    return sign_block(spec, state, block)
 
 
 def get_state_root(spec, state, slot) -> bytes: